Achieving Fully Proportional Representation: Approximability Results
aa r X i v : . [ c s . A I] D ec Achieving Fully Proportional Representation: ApproximabilityResults ∗ Piotr SkowronUniversity of WarsawWarsaw, Poland Piotr FaliszewskiAGH UniversityKrakow, Poland Arkadii SlinkoUniversity of AucklandAuckland, New Zealand
Abstract
We study the complexity of (approximate) winner determination under the Monroeand Chamberlin–Courant multiwinner voting rules, which determine the set of represen-tatives by optimizing the total (dis)satisfaction of the voters with their representatives.The total (dis)satisfaction is calculated either as the sum of individual (dis)satisfactions(the utilitarian case) or as the (dis)satisfaction of the worst off voter (the egalitariancase). We provide good approximation algorithms for the satisfaction-based utilitarianversions of the Monroe and Chamberlin–Courant rules, and inapproximability resultsfor the dissatisfaction-based utilitarian versions of them and also for all egalitariancases. Our algorithms are applicable and particularly appealing when voters submittruncated ballots. We provide experimental evaluation of the algorithms both on real-life preference-aggregation data and on synthetic data. These experiments show thatour simple and fast algorithms can in many cases find near-perfect solutions.
We study the complexity of (approximate) winner determination under the Monroe [32]and Chamberlin–Courant [10] multiwinner voting rules, which aim at selecting a group ofcandidates that best represent the voters. Multiwinner elections are important both forhuman societies (e.g., in indirect democracies for electing committees of representatives likeparliaments) and for software multiagent systems (e.g., for recommendation systems [25]),and thus it is important to have good multiwinner rules and good algorithms for them. TheMonroe and Chamberlin–Courant rules are particularly appealing because they create anexplicit (and, in some sense, optimal) connection between the elected committee membersand the voters; each voter knows his or her representative and each committee memberknows to whom he or she is accountable. In the context of recommendation systems this ∗ This paper combines and extends results presented at IJCAI-2013 (paper titled “Fully ProportionalRepresentation as Resource Allocation: Approximability Results”; the paper contained most of our theo-retical results) and at AAMAS-2012 (paper titled “Achieving Fully Proportional Representation is Easy inPractice”; the paper contained most of our experimental results). m candidates participate in the election and that the society consists of n voters, whoeach rank the candidates, expressing their preferences about who they would like to see astheir representative.When choosing a K -member committee, the Monroe and Chamberlin–Courant ruleswork as follows. For each voter they assign a single candidate as their representative,respecting the following rules:(a) altogether exactly K candidates are assigned to the voters. For the Monroe rule, eachcandidate is assigned either to about nK voters or to none; for the Chamberlin–Courantrule there is no such restriction and each committee member might be representinga different number of voters. The committee should take this into account in itsoperation, i.e., by means of weighted voting.(b) the candidates are selected and assigned to the voters optimally minimizing the total(societal) dissatisfaction or maximizing the total (societal) satisfaction.The total (dis)satisfaction is calculated on the basis of individual (dis)satisfactions. Weassume that there is a function α : N → N such that α ( i ) measures how well a voter is rep-resented by the candidate that this voter ranks as i ’th best. The function α is the same foreach voter. We can view α either as a satisfaction function (then it should be a decreasingone) or as a dissatisfaction function (then it should be an increasing one). For example, it istypical to use the Borda count scoring function whose m -candidate dissatisfaction variant isdefined as α m B , inc = i −
1, and whose satisfaction variant is α m B , dec = m − i . In the utilitarianvariants of the rules, the assignment should maximize (minimize) the total satisfaction (dis-satisfaction) calculated as the sum of the voters’ individual satisfactions (dissatisfactions)with their representatives. In the egalitarian variants, the assignment should maximize(minimize) the total satisfaction (dissatisfaction) calculated as the satisfaction (dissatisfac-tion) of the worst-off voter.The Monroe and Chamberlin–Courant rules create a useful connection between thevoters and their representatives that makes it possible to achieve both candidates’ account-ability to the voters, and proportional representation of voters’ views. Among commonvoting rules, the Monroe and Chamberlin–Courant rules seem to be unique in having both the accountability and the proportionality properties simultaneously. For example, FirstPast the Post system (where the voters are partitioned into districts with a separate single-winner Plurality election in each) can give very disproportionate results (forcing some of thevoters to be represented by candidates they dislike). On the other side of the spectrum arethe party-list systems, which achieve perfect proportionality. In those systems the votersvote for the parties, based on these votes each party receives some number of seats in theparliament, and then each party distributes the seats among its members (usually followinga publicly available list of the party’s candidates). This makes the elected candidates feelmore accountable to apparatchiks of their parties than to the voters. Somewhere between2he First Past the Post system and the party-list systems, we have the single transferablevote rule (STV), but for STV it is difficult to tell which candidate represents which voters.Unfortunately, the Monroe and Chamberlin–Courant rules have one crucial drawbackthat makes them impractical. It is NP-hard to tell who the winners are! Specifically, NP-hardness of winner determination under the Monroe and Chamberlin–Courant rules wasshown by Procaccia et al. [37] and by Lu and Boutilier [25]. Worse yet, the hardness holdseven if various natural parameters of the election are small [7]. Rare easy cases includethose, where the committee to be elected is small, or we consider the Chamberlin–Courantrule and the voters have single-peaked [7] or single-crossing preferences [43].Lu and Boutilier [25] proposed to use approximation algorithms and have given the firstsuch algorithm for the Chamberlin–Courant system. Their procedure outputs an assignmentthat achieves no less than 1 − e ≈ .
63 fraction of the optimal voter satisfaction. However,the approximation ratio 0 .
63 here means that it is possible that, on average, each agentis represented by a candidate that this agent prefers to only about 63% of the candidates,even if there is a perfect solution that assigns each agent to their most preferred candidate.Such issues, however, would not occurr if we had a constant-factor approximation algorithmminimizing the total dissatisfaction. Indeed, if a perfect solution exists, then the optimaldissatisfaction is zero and a constant-factor approximation algorithm must also output thisperfect solution.The use of approximation algorithms in real-life applications requires some discussion.For example, their use is naturally justified in the context of recommendation systems. Herethe strive for optimality is not crucial since a good but not optimal recommendation stillhas useful information and nobody would object if we replaced the exact recommendationwith an approximate one (given that the exact one is hard to calculate). For example,Amazon.com may recommend you a book on gardening which may not be the best bookfor you on this topic, but still full of useful advice. For such situations, Herbert Simon [41]used the term ‘satisficing,’ instead of optimizing, to explain the behavior of decision makersunder circumstances in which an optimal solution cannot be easily determined. On page129 he wrote: “Evidently, organisms adapt well enough to satisfice; they do not, in general,‘optimize’.” Effectively, what Simon says is that the use of approximation algorithms fitswell with the human nature.Still, the use of approximation algorithms in elections requires some care. It is con-ceivable that the electoral commission finds an allocation of voters to candidates with acertain value of (dis)satisfaction and one of the parties participating in the election finds anallocation with a better value. This can lead to a political deadlock. There are two ways ofavoiding this. Firstly, an approximation algorithm can be fixed by law. In such a case, itbecomes an acting voting rule and a new way to measure fairness in the society. Secondly,an electoral commission may calculate the allocation, but also publish the raw data andissue a call for submissions. If, within the period specified by law, nobody can producea better allocation, then the committee goes ahead and announces the result. If someoneproduces a better allocation, then the electoral commission uses the latter one.The use of approximation algorithms is even more natural in elections with partial3allots. Indeed, even if we use an exact algorithm to calculate the winners, the resultswill be approximate anyway since the voters provide us with approximations of their realpreferences and not with their exact preferences.
In this paper we focus on approximation algorithms for winner determination under theMonroe and Chamberlin–Courant rules. Our first goal is to seek algorithms that findassignments for which the dissatisfaction of voters is within a fixed bound of the optimalone. Unfortunately, we have shown that under standard complexity-theoretic assumptionssuch algorithms do not exist. Nonetheless, we found good algorithms that maximize voter’ssatisfaction. Specifically, we have obtained the following results:1. The Monroe and Chamberlin–Courant rules are hard to approximate up to any con-stant factor for the dissatisfaction-based cases (both utilitarian and egalitarian ones;see Theorems 1, 2, 3 and 4) and for the satisfaction-based egalitarian cases (see The-orems 5 and 7).2. For the satisfaction-based utilitarian framework we show the following. For the Mon-roe rule with the Borda scoring function we give a (0 . − ǫ )-approximation algorithm(often, the ratio is much better; see Section 4). In case of an arbitrary positionalscoring function we give a (1 − e )-approximation algorithm (Theorem 13). For theChamberlin–Courant rule with the Borda scoring function we give a polynomial-timeapproximation scheme (that is, for each ǫ , 0 < ǫ <
1, we have a polynomial-time(1 − ǫ )-approximation algorithm; see Theorem 15).3. We provide empirical evaluation of our algorithms for the satisfaction-based utilitarianframework, both on synthetic and real-life data. This evaluation shows that in practiceour best algorithms achieve at least 0 . A large number of papers are related to our research in terms of methodology (the study ofcomputational complexity and approximation algorithms for winner determination under4arious NP-hard election rules), in terms of perspective and motivation (e.g., due to theresource allocation view of Monroe and Chamberlin–Courant rules that we take), and interms of formal similarity (e.g., winner determination under the Chamberlin–Courant rulecan be seen as a form of the facility location problem). Below we review this relatedliterature.There are several single-winner voting rules for which winner determination is known tobe NP-hard. These rules include, for example, Dodgson’s rule [3,6,20], Young’s rule [6,38],and Kemeny’s rule [3,5,21]. For the single-transferable vote rule (STV), the winner determi-nation problem becomes NP-hard if we use the so-called parallel-universes tie-breaking [12].Many of these hardness results hold even in the sense of parameterized complexity theory(however, there also is a number of fixed-parameter tractability results; see the referencesabove for details).These hardness results motivated the search for approximation algorithms. There arenow very good approximation algorithms for Kemeny’s rule [1,13,24] and for Dodgson’srule [8,9,16,22,30]. In both cases the results are, in essence, optimal. For Kemeny’s rulethere is a polynomial-time approximation scheme [24] and for Dodgson’s rule the achievedapproximation ratio is optimal under standard complexity-theoretic assumptions [8] (un-fortunately, the approximation ratio is not constant but depends logarithmically on thenumber of candidates). On the other hand, for Young’s rule it is known that no goodapproximation algorithms exist [8].The work of Caragiannis et al. [9] and of Faliszewski et al. [16] on approximate winnerdetermination for Dodgson’s rule is particularly interesting from our perspective. In theformer, the authors advocate treating approximation algorithms for Dodgson’s rule as votingrules in their own right and design them to have desirable properties. In the latter, theauthors show that a well-established voting rule (so-called Maximin rule) is a reasonable(though not optimal) approximation of Dodgson’s rule. This perspective is important foranyone interested in using approximation algorithms for winner determination in elections(as might be the case for our algorithms for the Monroe and Chamberlin–Courant rules).The hardness of the winner determination problem for the Monroe and Chamberlin–Courant rules have been considered in several papers. Procaccia, Rosenschein and Zohar [37]were the first to show the hardness of these two rules for the case of a particular approval-style dissatisfaction function. Their results were complemented by Lu and Boutilier [25],Betzler, Slinko and Uhlmann [7], Yu, Chan, and Elkind [45], Skowron et al. [43], andSkowron and Faliszewski [42]. These are showing the hardness in case of the Borda dissat-isfaction function, obtain results on parameterized hardness of the two rules, and results onhardness (or easiness) for the cases where the profiles are single-peaked or single-crossing.Further, Lu and Boutilier [25] initiated the study of approximability for the Chamberlin–Courant rule (and were the first to use satisfaction-based framework). Specifically, theygave the (1 − e )-approximation algorithm for the Chamberlin–Courant rule. The motiva-tion of Lu and Boutilier was coming from the point of view of recommendation systemsand, in that sense, our view of the rules is quite similar to theirs.In this paper we take the view that the Monroe and Chamberlin–Courant rules are5pecial cases of the following resource allocation problem. The alternatives are shareableresources, each with a certain capacity defined as the maximal number of agents that mayshare this resource. Each agent has preferences over the resources and is interested ingetting exactly one. The goal is to select a predetermined number K of resources andto find an optimal allocation of these resources (see Section 2 for details). This provides aunified framework for the two rules and reveals the connection of proportional representationproblem to other resource allocation problems. In particular, it closely resembles multi-unitresource allocation with single-unit demand [40, Chapter 11] (see also the work of Chevaleyreet al. [11] for a survey of the most fundamental issues in the multiagent resource allocationtheory) and resource allocation with sharable indivisible goods [2,11]. Below, we point outother connections of the Monroe and Chamberlin–Courant rules to several other problems. Facility Location Problems.
In the facility location problem, there are n customers lo-cated in some area and an authority, say a city council, that wants to establish a fixednumber k of facilities to serve those customers. Customers incur certain costs (saytransportation costs) of using the facilities. Further, setting up a facility costs as well(and this cost may depend on the facility’s location). The problem is to find k loca-tions for the facilities that would minimize the total (societal) cost. If these facilitieshave infinite capacities and can serve any number of customers, then each customerwould use his/her most preferred (i.e., closest) facility and the problem is similar tofinding the Chamberlin–Courant assignment. If the capacities of the facilities are fi-nite and equal, the problem looks like finding an assignment in the Monroe rule. Anessential difference between the two problems are the setup costs and the distancemetric. The parameterized complexity of the Facility Location Problem was investi-gated in Fellows and Fornau [17]. The papers of Procaccia et al. [37] and of Betzleret al. [7] contain a brief discussion of the connection between the Facility LocationProblem and the winner determination problem under the Chamberlin–Courant rule. Group Activity Selection Problem.
In the group activity selection problem [14] wehave a group of agents (say, conference attendees) and a set of activities (say, optionsthat they have for a free afternoon such as a bus city tour or wine tasting). The agentsexpress preferences regarding the activities and organisers try to allocate agents to ac-tivities to maximise their total satisfaction. If there are m possible activities but only k must be chosen by organisers, then we are in the Chamberline-Courant framework,if all activities can take all agents, and in the Monroe framework, if all activities havethe same capacities. The difference is that those capacities may be different and alsothat in the Group Activity Selection Problem we may allow expression of more com-plicated preferences. For example, an agent may express the following preference “Ilike wine-tasting best provided that at most 10 people participate in it, and otherwiseI prefer a bus city tour provided that at least 15 people participate, and otherwiseI prefer to not take part in any activity”. The Group Activity Selection Problem ismore general than the winner determination in the Monroe and Chamberline-Courant6ules. Some hardness and easiness results for this problem were obtained in [14], butthe investigation of this problem has only started.The above connections show that, indeed, the complexity of winner determination underthe Monroe and Chamberlin–Courant voting rules are interesting, can lead to progress inseveral other directions, and may have impact on other applications of artificial intelligence. We first define basic notions such as preference orders and positional scoring rules. Then wepresent our Resource Allocation Problem in full generality and discuss which restrictionsof it correspond to the winner determination problem for the Monroe and Chamberlin–Courant voting rules. Finally, we briefly recall relevant notions regarding computationalcomplexity.
Preferences.
For each n ∈ N , by [ n ] we mean { , . . . , n } . We assume that there is aset N = [ n ] of agents and a set A = { a , . . . a m } of alternatives . Each alternative a ∈ A has the capacity cap a ∈ N , which gives the total number of agents that can be assigned toit. Further, each agent i has a preference order ≻ i over A , i.e., a strict linear order of theform a π (1) ≻ i a π (2) ≻ i · · · ≻ i a π ( m ) for some permutation π of [ m ]. For an alternative a , bypos i ( a ) we mean the position of a in the i ’th agent’s preference order. For example, if a is the most preferred alternative for i then pos i ( a ) = 1, and if a is the least preferred onethen pos i ( a ) = m . A collection V = ( ≻ , . . . , ≻ n ) of agents’ preference orders is called a preference profile .We will often include subsets of the alternatives in the descriptions of preference orders.For example, if A is the set of alternatives and B is some nonempty strict subset of A , thenby B ≻ A − B we mean that for the preference order ≻ all alternatives in B are preferredto those outside of B .A positional scoring function (PSF) is a function α m : [ m ] → N . A PSF α m is an increasing positional scoring function (IPSF) if for each i, j ∈ [ m ], if i < j then α m ( i ) <α m ( j ). Analogously, a PSF α m is a decreasing positional scoring function (DPSF) if foreach i, j ∈ [ m ], if i < j then α m ( i ) > α m ( j ).Intuitively, if β m is an IPSF then β m ( i ) can represent the dissatisfaction that an agentsuffers when assigned to an alternative that is ranked i ’th in his or her preference order.Thus, we assume that for each IPSF β m it holds that β m (1) = 0 (an agent is not dissatisfiedby her top alternative). Similarly, a DPSF γ m measures an agent’s satisfaction and weassume that for each DPSF γ m it holds that γ m ( m ) = 0 (an agent is completely notsatisfied being assigned his or her least desired alternative). Sometimes we write α insteadof α m , when it cannot lead to a confusion.We will often speak of families α of IPSFs (DPSFs) of the form α = ( α m ) m ∈ N , where α m is a PSF on [ m ], such that:1. For a family of IPSFs it holds that α m +1 ( i ) = α m ( i ) for all m ∈ N and i ∈ [ m ].7. For a family of DPSFs it holds that α m +1 ( i + 1) = α m ( i ) for all m ∈ N and i ∈ [ m ].In other words, we build our families of IPSFs (DPSFs) by appending (prepending) valuesto functions with smaller domains. To simplify notation, we will refer to such families ofIPSFs (DPSFs) as normal IPSFs (normal DPSFs). We assume that each function α m from afamily can be computed in polynomial time with respect to m . Indeed, we are particularlyinterested in the Borda families of IPSFs and DPSFs defined by α m B , inc ( i ) = i − α m B , dec ( i ) = m − i , respectively. Assignment functions. A K - assignment function is any function Φ : N → A , such that k Φ( N ) k ≤ K (that is, it matches agents to at most K alternatives), and such that for everyalternative a ∈ A we have that k Φ − ( a ) k ≤ cap a (i.e., the number of agents assigned to a does not exceed a ’s capacity cap a ).We will also consider partial assignment functions. A partial K -assignment function isdefined in the same way as a regular one, except that it may assign a null alternative, ⊥ ,to some of the agents. It is convenient to think that for each agent i we have pos i ( ⊥ ) = m .In general, it might be the case that a partial K -assignment function cannot be extendedto a regular one. This may happen, for example, if the partial assignment function uses K alternatives whose capacities sum to less than the total number of voters. However, in thecontext of Chamberlin–Courant and Monroe rules it is always possible to extend a partial K -assignment function to a regular one.Given a normal IPSF (DPSF) α , we may consider the following three functions, eachassigning a positive integer to any assignment Φ: ℓ α (Φ) = n X i =1 α (pos i (Φ( i ))) ,ℓ α ∞ (Φ) = max ni =1 α (pos i (Φ( i ))) ,ℓ α min (Φ) = min ni =1 α (pos i (Φ( i ))).These functions are built from individual dissatisfaction (satisfaction) functions, so thatthey can measure the quality of the assignment for the whole society. In the utilitarianframework the first one can be viewed as a total (societal) dissatisfaction function in theIPSF case and a total (societal) satisfaction function in the DPSF case. The second and thethird can be used, respectively, as a total dissatisfaction and satisfaction functions for IPSFand DPSF cases in the egalitarian framework. We will omit the word total if no confusionmay arise.For each subset of the alternatives S ⊆ A such that k S k ≤ K , we denote as Φ Sα thepartial K -assignment that assigns agents only to the alternatives from S and such thatΦ Sα maximizes the utilitarian satisfaction ℓ α (Φ Sα ). (We introduce this notation only for theutilitarian satisfaction-based setting because it is useful to express appropriate algorithmsfor this case; for other settings we have hardness results only and this notation would notbe useful.) 8 he Resource Allocation Problem. Let us now define the resource allocation problemthat forms the base of our study. This problem stipulates finding an optimal K -assignmentfunction, where the optimality is relative to one of the total dissatisfaction or satisfactionfunctions that we have just introduced. The former is to be minimized and the latter is tobe maximized. Definition 1.
Let α be a normal IPSF. An instance of α - DU-Assignment problem (i.e., ofthe disatisfaction-based utilitarian assignment problem) consists of a set of agents N = [ n ] ,a set of alternatives A = { a , . . . a m } , a preference profile V of the agents, and a sequence (cap a , . . . , cap a m ) of alternatives’ capacities. We ask for an assignment function Φ suchthat: (1) k Φ( N ) k ≤ K ; (2) k Φ − ( a ) k ≤ cap a for all a ∈ A ; and (3) ℓ α (Φ) is minimized. Problem α - SU-Assignment (the satisfaction-based utilitarian assignment problem) isdefined identically except that α is a normal DPSF and condition (3) is replaced with“(3 ′ ) ℓ α (Φ) is maximal.” If we replace ℓ α with ℓ α ∞ in α - DU-Assignment then we obtainproblem α - DE-Assignment , i.e., the dissatisfaction-based egalitarian variant. If we replace ℓ α with ℓ α min in α - SU-Assignment then we obtain problem α - SE-Assignment , i.e., thesatisfaction-based egalitarian variant.Our four problems can be viewed as generalizations of the winner determination problemfor the Monroe [32] and Chamberlin–Courant [10] multiwinner voting systems (see theintroduction for their definitions). To model the Monroe system, it suffices to set thecapacity of each alternative to be k N k K (for simplicity, throughout the paper we assume that K divides k N k ). We will refer to thus restricted variants of our problems as the Monroe variants. To represent the Chamberlin–Courant system, we set alternatives’ capacities to k N k . We will refer to the so-restricted variants of our problems as CC variants. Computational Issues . For many normal IPSFs α and, in particular, for the BordaIPSF, even the above-mentioned restricted versions of the Resource Allocation Problem,namely, α - DU-Monroe , α - DE-Monroe , α - DU-CC , and α - DE-CC are NP-complete [7,37] (the same holds for the satisfaction-based variants of the problems). Thus we seekapproximate solutions.
Definition 2.
Let r be a real number such that r ≥ ( < r ≤ ) and let α be a normal IPSF(a normal DPSF). An algorithm is an r -approximation algorithm for α - DU-Assignment problem (for α - SU-Assignment problem) if on each instance I it returns a feasible assign-ment Φ such that ℓ α (Φ) ≤ r · OPT (such that ℓ α (Φ) ≥ r · OPT ), where
OPT is the optimaltotal dissatisfaction (satisfaction) ℓ α (Φ OPT ) . We define r -approximation algorithms for the egalitarian variants analogously. Lu andBoutilier [25] gave a (1 − e )-approximation algorithm for the SU-CC family of problems.Throughout this paper, we will consider each of the
Monroe and CC variants of theproblem and for each we will either prove inapproximability with respect to any constant r In general, this assumption is not as innocent as it may seem. Often dealing with cases there K does notdivide k N k requires additional insights and care. However, for our algorithms and results, the assumptionsimiplifies notation and does not lead to obscuring any unexpected difficulties. Definition 3.
An instance I of Set-Cover consists of set U = [ n ] (called the ground set),family F = { F , F , . . . , F m } of subsets of U , and positive integer K . We ask if there existsa set I ⊆ [ m ] such that k I k ≤ K and S i ∈ I F i = U . Definition 4.
X3C is a variant of
Set-Cover where k U k is divisible by , each memberof F has exactly three elements, and K = k U k . Set-Cover remains NP-complete even if we restrict each member of U to be containedin at most two sets from F (it suffices to note that this restriction is satisfied by Vertex-Cover , which is a special case of
Set-Cover ). X3C remains NP-complete even if weadditionally assume that n is divisible by 2 and each member of U appears in at most 3sets from F [19].We will also use results from the theory of parameterized complexity developed byDowney and Fellows [15]. This theory allows to single out a particular parameter of theproblem, say k , and analyze its ‘contribution’ to the overall complexity of the problem. Ananalogue of the class P here is the class FPT which is the class of problems that can besolved in time f ( k ) n O (1) , where n is the size of the input instance, and f is some computablefunction (for a fixed k everything gets polynomial). Parameterized complexity theory alsooperates with classes W[1] ⊆ W[2] ⊆ · · · which are believed to form a hierarchy of classes of hard problems (combined, they are analogous to the class NP). It holds that FPT ⊆ W[1],but it seems unlikely that FPT = W[1], let alone FPT = W[2]. We point the reader to thebooks of Niedermeier [34] and Flum and Grohe [18] for detailed overviews of parametrizedcomplexity theory. Interestingly, while both
Set-Cover and
Vertex-Cover are NP-complete, the former is W[2]-complete and the latter belongs to FPT (see, e.g., the book ofNiedermeier [34] for these now-standard results and their history).
We now present our inapproximability results for the Monroe and Chamberlin–Courantrules. Specifically, we show that there are no constant-factor approximation algorithms forthe dissatisfaction-based variants of the rules (both utilitarian and egalitarian) and for thesatisfaction-based egalitarian ones.Naturally, these inapproximability results carry over to more general settings. For exam-ple, unless P = NP, there are no polynomial-time constant-factor approximation algorithmsfor the general dissatisfaction-based Resource Allocation Problem. On the other hand, ourresults do not preclude good satisfaction-based approximation algorithms for the utilitariancase and, indeed, in Section 4 we provide such algorithms.
Theorem 1.
For each normal IPSF α and each constant factor r > , there is nopolynomial-time r -approximation algorithm for α - DU-Monroe unless
P = NP . bb m f m + m ′ − m l Agent 1:Agent 2:Agent 3:Agent 4:Agent 5:Agent 6:Agent n : The alternatives from the second group nα (3) rm + m ′ = m + n · α (3) · rC − positions m f + nα (3) r Figure 1: The alignment of the positions in the preference orders of the agents. The positionsare numbered from the left to the right. The left wavy line shows the positions m f ( · ), eachno greater than 3. The right wavy line shows the positions m l ( · ), each higher than nr · α (3).The alternatives from A (positions of one such an alternative is illustrated with the circle)are placed only between the peripheral wavy lines. Each alternative from A is placed onthe left from the middle wavy line exactly 2 times, thus each such alternative is placed onthe left from the right dashed line no more than 2 times (exactly two times at the figure). Proof.
Let us fix a normal IPSF α and let us assume, aiming at getting a contradiction,that there is some constant r > r -approximation algorithm A for α - DU-Monroe .Let I be an instance of X3C with ground set U = [ n ] and family F = { F , F , . . . , F m } of 3-element subsets of U . Without loss of generality, we assume that n is divisible by both2 and 3 and that each member of U appears in at most 3 sets from F .Using I , we build instance I M of α - DU-Monroe as follows. We set N = U (that is, theelements of the ground set are the agents) and we set A = A ∪ A , where A = { a , . . . , a m } is a set of alternatives corresponding to the sets from the family F and A is a set ofdummy alternatives of cardinality k A k = n r · α (3), needed for the construction. We let m ′ = k A k and rename the alternatives in A so that A = { b , . . . , b m ′ } . We set K = n .We build agents’ preference orders using the following algorithm. For each j ∈ N , set M f ( j ) = { a i | j ∈ F i } and M l = { a i | j F i } . Set m f ( j ) = k M f ( j ) k and m l ( j ) = k M l ( j ) k .As the frequency of the elements from U is bounded by 3, we have m f ( j ) ≤
3. For eachagent j we set his or her preference order to be of the form M f ( j ) ≻ j A ≻ j M l ( j ), wherethe alternatives in M f ( j ) and M l ( j ) are ranked in an arbitrary way and the alternativesfrom A are placed at positions m f ( j ) + 1 , . . . , m f ( j ) + m ′ in the way described below (seeFigure 1 for a high-level illustration of the construction).We place the alternatives from A in the preference orders of the agents in such a way11hat for each alternative b i ∈ A there are at most two agents that rank b i among their nr · α (3) top alternatives. The following construction achieves this effect. If ( i + j ) mod n < b i is placed at one of the positions m f ( j ) + 1 , . . . , m f ( j ) + nr · α (3) in j ’spreference order. Otherwise, b i is placed at a position with index higher than m f ( j )+ nr · α (3)(and, thus, at a position higher than nr · α (3)). This construction can be implementedbecause for each agent j there are exactly m ′ · n = nr · α (3) alternatives b i , b i , b i nα (3) r suchthat ( i k + j ) mod n < A on I M . We will show that ℓ α (Φ) ≤ n · α (3) · r if and only if I is a yes -instance of X3C .( ⇐ ) If there exists a solution for I (i.e., an exact cover of U with n sets from F ), thenwe can easily show an assignment in which each agent j is assigned to an alternative fromthe top m f ( j ) positions of his or her preference order (namely, one that assigns each agent j to the alternative a i ∈ A that corresponds to the set F i , from the exact cover of U , thatcontains j ). Thus, for the optimal assignment Φ OPT it holds that ℓ α (Φ OPT ) ≤ α (3) · n . Inconsequence, A must return an assignment with the total dissatisfaction at most nr · α (3).( ⇒ ) Let us now consider the opposite direction. We assume that A found an assignmentΦ such that ℓ α (Φ) ≤ nr · α (3) and we will show that I is a yes -instance of X3C . Since werequire each alternative to be assigned to either 0 or 3 agents, if some alternative b i from A were assigned to some 3 agents, at least one of them would rank b i at a position worsethan nr · α (3). This would mean that ℓ α (Φ) ≥ nr · α (3) + 1. Analogously, no agent j can be assigned to an alternative that is placed at one of the m l ( j ) bottom positions of j ’s preference order. Thus, only the alternatives in A have agents assigned to them and,further, if agents x , y , z , are assigned to some a i ∈ A , then it holds that F i = { x, y, z } (we will call each set F i for which alternative a i is assigned to some agents x, y, z selected ).Since each agent is assigned to exactly one alternative, the selected sets are disjoint. Sincethe number of selected sets is K = n , it must be the case that the selected sets form anexact cover of U . Thus, I is a yes -instance of X3C .One may wonder if hardness of approximation for α - DU-Monroe is not an artifactof the strict requirements regarding the number of chosen candidates. It turns out thatunless P = NP, there is no r - s -approximation algorithm that finds an assignment with thefollowing properties: (1) the aggregated dissatisfaction ℓ α (Φ) is at most r times higher thanthe optimal one, (2) the number of alternatives to which agents are assigned is at most sK and (3) each selected alternative (the alternative that has agents assigned), is assigned tono more than s ⌈ nK ⌉ and no less than s ⌈ nK ⌉ agents. (The proof is similar to the one used forTheorem 1.) Thus, in our further study we do not consider such relaxations of the problem. Theorem 2.
For each normal IPSF α and each constant r > , there is no polynomial-time r -approximation algorithm for α - DE-Monroe unless
P = NP .Proof.
The proof of Theorem 1 applies to this case as well. In fact, it even suffices to take m ′ = k A k = nr · α (3).Results analogous to Theorems 1 and 2 hold for the DU-CC family of problems as well.12 heorem 3.
For each normal IPSF α and each constant factor r > , there is nopolynomial-time r -approximation algorithm for α - DU-CC unless
P = NP .Proof.
Let us fix a normal IPSF α . For the sake of contradiction, let us assume that thereis some constant r >
1, and a polynomial-time r -approximation algorithm A for α - DU-CC . We will show that it is possible to use A to solve the NP-complete Vertex-Cover problem.Let I = ( U, F , K ) be an instance of Vertex-Cover , where U = [ n ] is the ground set, F = { F , . . . , F m } is a family of subsets of U (where each member of U belongs to exactlytwo sets in F ), and K is a positive integer.Given I , we construct an instance I CC of α - DU-CC as follows. The set of agentsis N = U and the set of alternatives is A = S mj =1 A j , where each A j contains exactly α (2) · r · n (unique) alternatives. Intuitively, for each j ∈ [ m ], the alternatives in A j correspond to the set F j . For each A j , 1 ≤ j ≤ m , we pick one alternative, which wedenote a j . For each agent i ∈ N , we set i ’s preference order as follows: Let F j and F k , j < k , be the two sets that contain i . Agent i ’s preference order is of the form a j ≻ i a k ≻ i A k − { a k } ≻ i A − ( A k ∪ { a j , a k } ) (a particular order of alternatives in the sets A k − { a k } and A − ( A k ∪ { a j , a k } ) is irrelevant for the construction). We ask for an assignment of theagents to at most K alternatives.Let us consider a solution Φ returned by A on input I CC . We claim that ℓ α (Φ) ≤ nr · α (2)if and only if I is a yes -instance of Vertex-Cover .( ⇐ ) If I is a yes -instance then, clearly, each agent i can be assigned to one of the top twoalternatives in his or her preference order (if there is a size- K cover, then this assignmentselects at most K candidates). Thus the total dissatisfaction of an optimal assignment isat most n · α (2). As a result, the solution Φ returned by A has total dissatisfaction at most nr · α (2).( ⇒ ) If A returns an assignment with total dissatisfaction no greater than nr · α (2), then,by the construction of agents preference orders, we see that each agent i was assigned toan alternative from a set A j such that i ∈ F j . Since the assignment can use at most K alternatives, this directly implies that there is a size- K cover of U with sets from F . Theorem 4.
For each normal IPSF α and each constant factor r > , there is nopolynomial-time r -approximation algorithm for α - DE-CC unless
P = NP .Proof.
The proof of Theorem 3 is applicable in this case as well. In fact, it even suffices totake the m groups of alternatives, A , . . . , A m , to contain α (2) · r alternatives each.The above results show that approximating algorithms for finding the minimal dissat-isfaction of agents is difficult. On the other hand, if we focus on agents’ total satisfactionthen constant-factor approximation exist in many cases (see, e.g., the work of Lu andBoutilier [25] and the next section). Yet, if we focus on the satisfaction of the least satis-fied voter, there are no efficient constant-factor approximation algorithms for the Monroeand Chamberlin–Courant systems. (However, note that our result for the Monroe setting13s more general than the result for the Chamberlin–Courant setting; the latter is for theBorda DPSF only.) Theorem 5.
For each normal DPSF α (where each entry is polynomially bounded inthe number of alternatives) and each constant factor r , with < r ≤ , there is no r -approximation algorithm for α - SE-Monroe unless
P = NP .Proof.
Let us fix a DPSF α = ( α m ) m ∈ N , where each entry α m is polynomially bounded inthe number of alternatives m . For the sake of contradiction, let us assume that for some r ,0 < r ≤
1, there is a polynomial-time r -approximation algorithm A for α - SE-Monroe . Wewill show that the existence of this algorithm implies that
X3C is solvable in polynomialtime.Let I be an X3C instance with ground set U = { , , . . . , n } and collection F = { F , . . . , F m } of subsets of U . Each set in F has cardinality three. Further, without lossof generality, we can assume that n is divisible by three and that each i ∈ U appears inat most three sets from F . Given I , we form an instance I M of α - SE-Monroe as follows.Let n ′ = 3 · ( α m +1 (1) · ⌈ − rr ⌉ + 3). The set N of agents is partitioned into two subsets, N and N . N contains n agents (intuitively, corresponding to the elements of the ground set U ) and N contains n ′ agents (used to enforce certain properties of the solution). The setof alternatives A is partitioned into two subsets, A and A . We set A = { a , . . . , a m } (members of A correspond to the sets in F ), and we set A = { b , . . . , b m ′ } , where m ′ = n ′ .For each j , 1 ≤ j ≤ n , we set M f ( j ) = { a i | j ∈ F i } . For each j , 1 ≤ j ≤ n , we set thepreference order of the j ’th agent in N to be of the form M f ( j ) ≻ A ≻ A − M f ( j ) . Note that by our assumptions, k M f ( j ) k ≤
3. For each j , 1 ≤ j ≤ n ′ , we set the preferenceorder of the j ’th agent in N to be of the form b ⌈ j ⌉ ≻ A − { b ⌈ j ⌉} ≻ A . Note that each agent in N ranks the alternatives from A in positions m ′ + 1 , . . . , m ′ + m .Finally, we set the number of candidates that can be selected to be K = n + n ′ .Now, consider the solution Φ returned by A on I M . We will show that ℓ α m + m ′ ∞ (Φ) ≤ rα m + m ′ (3) if and only if I is a yes -instance of X3C .( ⇐ ) If there exists an exact set cover of U with sets from F , then it is easy to constructa solution for I M where the satisfaction of each agent is greater or equal to r · α m + m ′ (3).Let I ⊆ { , . . . , m } be a set such that S i ∈ I F i = U and k I k = n . We assign each agent j from N to the alternative a i such that (a) i ∈ I and (b) j ∈ F i , and we assign eachagent from N to his or her most preferred alternative. Thus, Algorithm A has to returnan assignment with the minimal satisfaction greater or equal to r · α m + m ′ (3).( ⇒ ) For the other direction, we first show that r · α m + m ′ (3) ≥ α m + m ′ ( m ′ ). Since DPSFsare strictly decreasing, it holds that: r · α m + m ′ (3) ≥ r · ( α m + m ′ ( m ′ ) + m ′ − . (1)14hen, by the definition of DPSFs, it holds that: α m + m ′ ( m ′ ) = α m +1 (1) . (2)Using the fact that m ′ = ( α m +1 (1) ·⌈ − rr ⌉ +3) and using (2), we can transform inequality (1)to obtain the following: r · α m + m ′ (3) ≥ r · ( α m + m ′ ( m ′ ) + m ′ − r · (cid:18) α m + m ′ ( m ′ ) + ( α m +1 (1) · (cid:24) − rr (cid:25) + 3) − (cid:19) ≥ r · α m + m ′ ( m ′ ) + (1 − r ) · α m +1 (1)= r · α m + m ′ ( m ′ ) + (1 − r ) · α m + m ′ ( m ′ ) = α m + m ′ ( m ′ ) . This means that if the minimal satisfaction of an agent is at least r · α m + m ′ (3), then noagent was assigned to an alternative that he or she ranked beyond position m ′ . If someagent j from N were assigned to an alternative from A , then, by the pigeonhole principle,some agent from N would be assigned to an alternative from A . However, each agentin N ranks the alternatives from A beyond position m ′ and thus such an assignment isimpossible. In consequence, it must be that each agent in j was assigned to an alternativethat corresponds to a set F i in F that contains j . Such an assignment directly leads to asolution for I .Let us now move on to the case of SE-CC family of problems. Unfortunately, in thiscase our inapproximability argument holds for the case of Borda DPSF only (though webelieve that it can be adapted to other DPSFs as well). Further, in our previous theoremswe were showing that existence of a respective constant-factor approximation algorithmimplies that NP collapses to P. In the following theorem we will show a seemingly weakercollapse of W[2] to FPT.To prove hardness of approximation for α B , dec - SE-CC , we first prove the followingsimple lemma.
Lemma 6.
Let
K, p, l be three positive integers and let X be a set of cardinality lpK .There exists a family S = { S , . . . , S ( lKK ) } of pK -element subsets of X such that for each K -element subset B of X , there is a set S i ∈ S such that B ⊆ S i .Proof. Set X ′ = [ lK ] and let Y ′ be a family of all K -element subsets of X ′ . Replace eachelement i of X ′ with p new elements (at the same time replacing i with the same p elementswithin each set in Y ′ that contains i ). As a result we obtain two new sets, X and Y , thatsatisfy the statement of the theorem (up to the renaming of the elements). Theorem 7.
Let α m B , dec be the Borda DPSF ( α m B , dec ( i ) = m − i ). For each constant factor r , < r ≤ , there is no r -approximation algorithm for α m B , dec - SE-CC unless
FPT = W[2] . roof. For the sake of contradiction, let us assume that there is some constant r , 0 < r ≤ r -approximation algorithm A for α m B , dec - SE-CC . We will show thatthe existence of this algorithm implies that
Set-Cover is fixed-parameter tractable for theparameter K (since Set-Cover is known to be W[2]-complete for this parameter, this willimply FPT = W[2]).Let I be an instance of Set-Cover with ground set U = [ n ] and family F = { F , F , . . . , F m } of subsets of U . Given I , we build an instance I CC of α m B , dec - SE-CC as follows. The set of agents N consists of n subsets of agents, N , . . . , N n , where eachgroup N i contains exactly n ′ = (cid:0) ⌈ r ⌉ KK (cid:1) agents. Intuitively, for each i , 1 ≤ i ≤ n , the agentsin the set N i correspond to the element i in U . The set of alternatives A is partitionedinto two subsets, A and A , such that: (1) A = { a , . . . , a m } is a set of alternatives cor-responding to the sets from the family F , and (2) A , k A k = (cid:6) r (cid:7) l m (1+ r ) K m K , is a set ofdummy alternatives needed for our construction. We set m ′ = k A k = m + k A k .Before we describe the preference orders of the agents in N , we form a family R = { r , . . . , r n ′ } of preference orders over A that satisfies the following condition: For each K -element subset B of A , there exists r j in R such that all members of B are rankedamong the bottom l m (1+ r ) K m K positions in r j . By Lemma 6, such a construction is possible(it suffices to take l = (cid:6) r (cid:7) and p = l m (1+ r ) K m ); further, the proof of the lemma provides analgorithmic way to construct R .We form the preference orders of the agents as follows. For each i , 1 ≤ i ≤ n , set M f ( i ) = { a t | i ∈ F t } . For each i , 1 ≤ i ≤ n , and each j , 1 ≤ j ≤ n ′ , the j ’th agent from N i has preference order of the form: M f ( i ) ≻ r j ≻ A − M f ( i )(we pick any arbitrary, polynomial-time computable order of candidates within M f ( i ) and M l ( i )).Let Φ be an assignment computed by A on I M . We will show that ℓ α m ′ B , dec ∞ (Φ) ≥ r · ( m ′ − m )if and only if I is a yes -instance of Set-Cover .( ⇐ ) If there exists a solution for I (i.e., a cover of U with K sets from F ), then wecan easily show an assignment where each agent is assigned to an alternative that he or sheranks among the top m positions (namely, for each j , 1 ≤ j ≤ n , we assign all the agentsfrom the set N j to the alternative a i ∈ A such that j ∈ F i and F i belongs to the alleged K -element cover of U ). Under this assignment, the least satisfied agent’s satisfaction is atleast m ′ − m and, thus, A has to return an assignment Φ where ℓ α m ′ B , dec ∞ (Φ) ≥ r · ( m ′ − m ).( ⇒ ) Let us now consider the opposite direction. We assume that A found an assignmentΦ such that ℓ α m B , dec ∞ (Φ) ≥ r · ( m ′ − m ) and we will show that I is a yes -instance of Set-Cover . We claim that for each i , 1 ≤ i ≤ n , at least one agent j in N i were assigned to analternative from A . If all the agents in N i were assigned to alternatives from A , then, bythe construction of R , at least one of them would have been assigned to an alternative that he16r she ranks at a position greater than k A k − l m (1+ r ) K m K = (cid:6) r (cid:7) l m (1+ r ) K m K − l m (1+ r ) K m K .For x = l m (1+ r ) K m K we have: (cid:24) r (cid:25) x − x ≥ m ′ − m ′ r + mr (we skip the straightforward calculation) and, thus, this agent would have been assignedto an alternative that he or she ranks at a position greater than m ′ − m ′ r + mr . As aconsequence, this agent’s satisfaction would be lower than ( m ′ − m ) r . Similarly, no agentfrom N i can be assigned to an alternative from M l ( i ). Thus, for each i , 1 ≤ i ≤ n ,there exists at least one agent j ∈ N i that is assigned to an alternative from M f ( i ). Inconsequence, the covering subfamily of F consists simply of those sets F k , for which someagent is assigned to alternative a k ∈ A .The presented construction gives the exact algorithm for Set-Cover problem runningin time f ( K )( n + m ) O (1) , where f ( K ) is polynomial in (cid:0) ⌈ r ⌉ K (cid:1) . The existence of such analgorithm means that Set-Cover is in FPT. On the other hand, we know that
Set-Cover is W[2]-complete, and thus if A existed then FPT = W[2] would hold. We now turn to approximation algorithms for the Monroe and Chamberlin–Courant mul-tiwinner voting rules in the satisfaction-based framework. Indeed, if one focuses on agents’total satisfaction then it is possible to obtain high-quality approximation results. In par-ticular, we show the first nontrivial (randomized) approximation algorithm for α B , dec - SU-Monroe . We show that for each ǫ > . − ǫ approximation ratio; the algorithm usually gives evenbetter approximation guarantees. For the case of arbitrarily selected DPSF we show a(1 − e − )-approximation algorithm. Finally, we show the first polynomial-time approxima-tion scheme (PTAS) for α B , dec - SU-CC . These results stand in sharp contrast to those fromthe previous section, where we have shown that approximation is hard for essentially allremaining variants of the problem.The core difficulty in solving α - Monroe/CC-Assignment problems lays in selectingthe alternatives that should be assigned to the agents. Given a preference profile and aset S of up to K alternatives, using a standard network-flow argument, it is easy to find a(possibly partial) optimal assignment Φ Sα of the agents to the alternatives from S . Proposition 8 ( Implicit in the paper of Betzler et al. [7] ) . Let α be a normal DPSF, N be a set of agents, A be a set of alternatives (togehter with their capacities; perhapsrepresented implicitly as for the case of the Monroe and Chamberlin–Courant rules), V be apreference profile of N over A , and S a K -element subset of A (where K divides k N k ). Thenthere is a polynomial-time algorithm that computes a (possibly partial) optimal assignment Φ Sα of the agents to the alternatives from S . otation : Φ ← a map defining a partial assignment, iteratively built by the algorithm.Φ ← ← the set of agents for which the assignment is already defined.Φ → ← the set of alternatives already used in the assignment. if K ≤ then compute the optimal solution using an algorithm of Betzler et al. [7] and return.Φ = {} for i ← to K do score ← {} bests ← {} foreach a i ∈ A \ Φ → do agents ← sort N \ Φ ← so that if agent j precedes agent k then pos j ( a i ) ≤ pos k ( a i ) bests [ a i ] ← chose first ⌈ NK ⌉ elements from agentsscore [ a i ] ← P j ∈ bests [ a i ] ( m − pos j ( a i )) a best ← argmax a ∈ A \ Φ → score [ a ] foreach j ∈ bests [ a best ] do Φ[ j ] ← a best Figure 2: The pseudocode for Algorithm A.Note that for the case of the Chamberlin–Courant rule the algorithm from the aboveproposition can be greatly simplified: To each voter we assign the candidate that he or sheranks highest among those from S . For the case of Monroe, unfortunately, we need theexpensive network-flow-based approach. Nonetheless, Proposition 8 allows us to focus onthe issue of selecting the winning alternatives and not on the issue of matching them to theagents.Below we describe our algorithms for α B , dec - SU-Monroe and for α B , dec - SU-CC . For-mally speaking, every approximation algorithm for α B , dec - SU-Monroe also gives feasibleresults for α B , dec - SU-CC . However, some of our algorithms are particularly well-suited forboth problems and some are tailored to only one of them. Thus, for each algorithm weclearly indicate if it is meant only for the case of Monroe, only for the case of CC, or if itnaturally works for both systems.
Perhaps the most natural approach to solve α B , dec - SU-Monroe is to build a solutioniteratively: In each step we pick some not-yet-assigned alternative a i (using some criterion)and assign it to those ⌈ NK ⌉ agents that (a) are not assigned to any other alternative yet,and (b) whose satisfaction of being matched with a i is maximal. It turns out that this idea,implemented formally as Algorithm A (see pseudo code in Figure 2), works very well inmany cases. We provide a lower bound on the total satisfaction it guarantees in the nextlemma. We remind the reader that the so-called k ’th harmonic number H k = P ki =1 1 i hasasymptotics H k = Θ(log k ). Lemma 9.
Algorithm A is a polynomial-time (1 − K − m − − H K K ) -approximation algorithm or α B , dec - SU-Monroe .Proof.
Our algorithm explicitly computes an optimal solution when K ≤ K ≥
3. Let us consider the situation in the algorithm after the i ’th iteration of theouter loop (we have i = 0 if no iteration has been executed yet). So far, the algorithm haspicked i alternatives and assigned them to i nK agents (recall that for simplicity we assumethat K divides n evenly). Hence, each agent has ⌈ m − iK − i ⌉ unassigned alternatives among hisor her i + ⌈ m − iK − i ⌉ top-ranked alternatives. By pigeonhole principle, this means that thereis an unassigned alternative a ℓ who is ranked among top i + ⌈ m − iK − i ⌉ positions by at least nK agents. To see this, note that there are ( n − i nK ) ⌈ m − iK − i ⌉ slots for unassigned alternativesamong the top i + ⌈ m − iK − i ⌉ positions in the preference orders of unassigned agents, and thatthere are m − i unassigned alternatives. As a result, there must be an alternative a ℓ forwhom the number of agents that rank him or her among the top i + ⌈ m − iK − i ⌉ positions is atleast: 1 m − i (cid:18) ( n − i nK ) (cid:24) m − iK − i (cid:25)(cid:19) ≥ nm − i (cid:18) K − iK (cid:19) (cid:18) m − iK − i (cid:19) = nK . In consequence, the ⌈ nK ⌉ agents assigned in the next step of the algorithm will have thetotal satisfaction at least ⌈ nK ⌉ · ( m − i − ⌈ m − iK − i ⌉ ). Thus, summing over the K iterations, thetotal satisfaction guaranteed by the assignment Φ computed by Algorithm Ais at least thefollowing value: (to derive the fifth line from the fourth one we note that K ( H K − − H K ≥ K ≥ ℓ α b (Φ) ≥ K − X i =0 nK · (cid:18) m − i − ⌈ m − iK − i ⌉ (cid:19) ≥ K − X i =0 nK · (cid:18) m − i − m − iK − i − (cid:19) = K X i =1 nK · (cid:18) m − i − m − K − i + 1 + i − K − i + 1 (cid:19) = nK (cid:18) K (2 m − K − − ( m − H K + K ( H K − − H K (cid:19) ≥ nK (cid:18) K (2 m − K − − ( m − H K (cid:19) ≥ ( m − n (cid:18) − K − m − − H K K (cid:19) If each agent were assigned to his or her top alternative, the total satisfaction would beequal to ( m − n . Thus we get the following bound: ℓ α B , dec (Φ)OPT ≤ − K − m − − H K K .
This completes the proof. 19ote that in the above proof we measure the quality of our assignment against, aperhaps-impossible, perfect solution, where each agent is assigned to his or her top al-ternative. This means that for relatively large m and K , and small Km ratio, the algorithmcan achieve a close-to-ideal solution irrespective of the voters’ preference orders. We believethat this is an argument in favor of using Monroe’s system in multiwinner elections. Onthe flip side, to obtain a better approximation ratio, we would have to use a more involvedbound on the quality of the optimal solution. To see that this is the case, form an instance I of α B , dec - SU-Monroe with n agents and m alternatives, where all the agents have thesame preference order, and where we seek to elect K candidates (and where K divides n ).It is easy to see that each solution that assigns the K universally top-ranked alternatives tothe agents is optimal. Thus the total dissatisfaction of the agents in the optimal solutionis: nK (( m −
1) + · · · + ( m − K )) = nK (cid:18) K (2 m − K − (cid:19) = n ( m − (cid:18) − K − m − (cid:19) . By taking large enough m and K (even for a fixed value of mK ), the fraction 1 − K − m − can be arbitrarily close to the approximation ratio of our algorithm (the reasoning here issomewhat in the spirit of the idea of identifying maximally robust elections, as studied byShiryaev, Yu, and Elkind [39]).For small values of K , it is possible that the H K K part of our approximation ratio woulddominate the K − m − part. In such cases we can use the result of Betzler et al. [7], whoshowed that for each fixed constant K , α B , dec - SU-Monroe can be solved in polynomialtime. Thus, for the finite number of cases where H K K is too large, we can solve the problemoptimally using their algorithm. In consequence, the quality of the solution produced byAlgorithm A most strongly depends on the ratio K − m − . In most cases we can expect it tobe small (for example, in Polish parliamentary elections K = 460 and m ≈ . m and K in Figure 3. Proposition 10.
Let P be the number of top positions in the agents’ preference orders thatare known by the algorithm. In this case Algorithm A is a polynomial-time r -approximationalgorithm for α B , dec - SU-Monroe , where: r = K − X i =0 n ( m −
1) max( r ( i ) ,
0) 20 nd r ( i ) = nK ( m − i − m − iK − i ) if (cid:16) i + m − iK − i (cid:17) ≤ P , nK ( K − i )( m − i )4 if (cid:16) i + m − iK − i (cid:17) > P and (2 P − m ) ≥ i ≥ ( K − , nK ( m − P )( K − i )( P − i ) m − i otherwise.Proof. We use the same approach as in the proof of Lemma 9, except that we adjustour estimates of voters’ satisfaction. Consider a situation after some i ’th iteration of thealgorithm’s outer loop ( i = 0 if we are before the first iteration). If i + m − iK − i ≤ P , then wecan use the same lower bound for the satisfaction of the agents assigned in the ( i + 1)’thiteration as in the proof of Lemma 9. That is, the agents assigned in the ( i + 1)’th iterationwill have satisfaction at least r ( i ) = nK · ( m − i − m − iK − i ).For the case where i + m − iK − i > P , the bound from Lemma 9 does not hold, but we canuse a similar approach to find a different one. Let P x ≤ P be some positive integer. Weare interested in the number x of not-yet assigned agents who rank some not-yet-selectedalternative among their top P x positions (after the i ’th iteration). Similarly as in the proofof Lemma 9, using the pigeonhole principle we note that: x ≥ m − i (cid:16) n − i nK (cid:17) ( P x − i ) = nK · ( K − i )( P x − i ) m − i .Thus, the satisfaction of the agents assigned in the ( i + 1)’th iteration is at least:min (cid:16) x, nK (cid:17) ( m − P x ) = nK · ( m − P x ) min (cid:18) ( K − i )( P x − i ) m − i , (cid:19) . (3)The case ( K − i )( P x − i ) m − i ≥ i + m − iK − i ≤ P x ) implies that i + m − iK − i ≤ P and forthis case we lower-bound agents’ satisfaction by r ( i ). For the case where ( K − i )( P x − i ) m − i ≤ i + m − iK − i ≥ P x , equation (3) simplifies to: nK · ( m − P x ) · ( K − i )( P x − i ) m − i . (4)We use this estimate for the satisfaction of the agents assigned in the ( i + 1)’th iterationfor the cases where (a) i + m − iK − i ≥ m + i and (b) m + i ≤ P (or, equivalently, (2 P − m ) ≥ i ≥ ( K − nK · ( m − P x ) · ( K − i )( P x − i ) m − i ≥ nK · ( m − m + i · ( K − i )( m + i − i ) m − i = nK · ( K − i )( m − i ) m − i ) = nK · ( K − i )( m − i )4 .21or the remaining cases, we set P x = P and (4) becomes: nK · ( m − P )( K − i )( P − i ) m − i .Naturally, we replace our estimates by 0 whenever they become negative.To complete the proof, it suffice to, as in the proof of Lemma 9, note that ( m − n isan upper bound on the satisfaction achieved by the optimal solution. a pp r o x i m a t i o n g u a r a n t ee K =5, m =100)( K =10, m =100)( K =20, m =100)( K =30, m =100) 00.20.40.60.81 a pp r o x i m a t i o n g u a r a n t ee K =50, m =1000)( K =100, m =1000)( K =200, m =1000)( K =300, m =1000) Figure 3: The relation between the percentage of the known positions and the approximationratio of Algorithm A for α B , dec - SU-Monroe .For example, for the case of Polish parliamentary elections ( K = 460 and m = 6000),to achieve 90% of voters’ optimal satisfaction, each voter would have to rank about 8 . There are simple ways in which we can improve the quality of the assignments producedby Algorithm A. For example, our Algorithm B first runs Algorithm A and then, usingProposition 8, optimally reassigns the alternatives to the voters. As shown in Section 5, thisvery simple trick turns out to noticeably improve the results of the algorithm in practice(and, of course, the theoretical approximation guarantees of Algorithm A carry over toAlgorithm B).
Algorithm C is a further heuristic improvement over Algorithm B. This time the idea isthat instead of keeping only one partial function Φ that is iteratively extended up to the full22 otation : We use the same notation as in Algorithm A;
P ar ← a list of partial representation functions P ar = []
P ar .push( {} ) for i ← to K do newP ar = [] for Φ ∈ P ar do bests ← {} foreach a i ∈ A \ Φ → do agents ← sort N \ Φ ← (agent j precedes agent k implies that pos j ( a i ) ≤ pos k ( a i )) bests [ a i ] ← chose first ⌈ NK ⌉ elements of agents Φ ′ ← Φ foreach j ∈ bests [ a i ] do Φ ′ [ j ] ← a i newP ar .push(Φ ′ )sort newP ar according to descending order of the total satisfaction of the assignedagents P ar ← chose first d elements of newP ar for Φ ∈ P ar do Φ ← compute the optimal representative function using an algorithm of Betzler et al. [7]for the set of winners Φ → return the best representative function from P ar
Figure 4: The pseudocode for Algorithm C.assignment, we keep a list of up to d partial assignment functions, where d is a parameter ofthe algorithm. At each iteration, for each assignment function Φ among the d stored onesand for each alternative a that does not yet have agents assigned to by this Φ, we computean optimal extension of this Φ that assigns agents to a . As a result we obtain possibly morethan d (partial) assignment functions. For the next iteration we keep those d of them thatgive highest satisfaction.We provide pseudocode for Algorithm C in Figure 4. If we take d = 1, we obtainAlgorithm B. If we also disregard the last two lines prior to returning the solution, weobtain Algorithm A.Algorithm C can also be adapted for the Chamberlin–Courant rule. The only differenceconcerns creating the assignment functions: we replace the contents of the first foreach loopwith the following code: foreach a i ∈ A \ Φ → do Φ ′ ← Φ foreach j ∈ N doif agent j prefers a i to Φ ′ ( j ) then Φ ′ ( j ) ← a i newP ar .push(Φ ′ ) Note that, for the case of the Chamberlin–Courant rule, Algorithm C can also be seen as ageneralization of Algorithm GM that we will discuss later in Section 4.5.23 .4 Algorithm R (Monroe, CC)
Algorithms A, B and C achieve very high approximation ratios for the cases where K issmall relative to m . For the remaining cases, where K and m are comparable, we can usea sampling-based randomized algorithm (denoted as Algorithm R) described below. Wefocus on the case of Monroe and we will briefly mention the case of CC at the end.The idea of this algorithm is to randomly pick K alternatives and match them optimallyto the agents, using Proposition 8. Naturally, such an algorithm might be very unlucky andpick K alternatives that all of the agents rank low. Yet, if K is comparable to m then it islikely that such a random sample would include a large chunk of some optimal solution. Inthe lemma below, we asses the expected satisfaction obtained with a single sampling step(relative to the satisfaction given by the optimal solution) and the probability that a singlesampling step gives satisfaction close to the expected one. Naturally, in practice one shouldtry several sampling steps and pick the one with the highest satisfaction. Lemma 11.
A single sampling step of the randomized algorithm for α B , dec - SU-Monroe achieves expected approximation ratio of (1 + Km − K m − m + K m − m ) . Let p ǫ denote theprobability that the relative deviation between the obtained total satisfaction and the expectedtotal satisfaction is higher than ǫ . Then for K ≥ we have p ǫ ≤ exp (cid:16) − Kǫ (cid:17) .Proof. Let N = [ n ] be the set of agents, A = { a , . . . , a m } be the set of alternatives, and V be the preference profile of the agents. Let us fix some optimal solution Φ opt and let A opt bethe set of alternatives assigned to the agents in this solution. For each a i ∈ A opt , we writesat( a i ) to denote the total satisfaction of the agents assigned to a i in Φ opt . Naturally, wehave P a ∈ A opt sat( a ) = OPT. In a single sampling step, we choose uniformly at random a K -element subset B of A . Then, we form a solution Φ B by matching the alternatives in B optimally to the agents (via Proposition 8). We write K opt to denote the random variableequal to k A opt ∩ B k , the number of sampled alternatives that belong to A opt . We define p i = Pr( K opt = i ). For each j ∈ { , . . . , K } , we write X j to denote the random variableequal to the total satisfaction of the agents assigned to the j ’th alternative from the sample.We claim that for each i , 0 ≤ i ≤ K , it holds that: E K X j =1 X j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K opt = i ≥ iK OPT + m − i − · (cid:16) n − i nK (cid:17) . Why is this so? Given a sample B that contains i members of A opt , our algorithm’s solutionis at least as good as a solution that matches the alternatives from B ∩ A opt in the same wayas Φ opt , and the alternatives from B − A opt in a random manner. Since K opt = i and each a j ∈ A opt has equal probability of being in the sample, it is easy to see that the expectedvalue of P a j ∈ B ∩ A opt sat( a j ) is iK OPT. After we allocate the agents from B ∩ A opt , each ofthe remaining, unassigned agents has m − i positions in his or her preference order wherehe ranks the agents from A − A opt . For each unassigned agents, the average score valueassociated with these positions is at least m − i − (this is so, because in the worst case the24gent could rank the alternatives from B ∩ A opt in the top i positions). There are ( n − i nK )such not yet assigned agents and so the expected total satisfaction from assigning themrandomly to the alternatives is m − i − · ( n − i nK ). This proves our bound on the expectedsatisfaction of a solution yielded by optimally matching a random sample of K alternatives.Since OPT is upper bounded by ( m − n (consider a possibly-nonexistent solutionwhere every agent is assigned to his or her top preference), we get that: E K X j =1 X j | K opt = i ≥ iK OPT + m − i − m − · (cid:18) − iK (cid:19) OPT . We can compute the unconditional expected satisfaction of Φ B as follows: E K X j =1 X j = K X i =0 p i E K X j =1 X j | K opt = i ≥ K X i =0 p i (cid:18) iK OPT + m − i − m − · (cid:18) − iK (cid:19) OPT (cid:19) . Since P Ki =1 p i · i is the expected number of the alternatives in A opt , we have that P Ki =1 p i · i = K m (one can think of summing the expected values of K indicator random variables; one foreach element of A opt , taking the value 1 if a given alternative is selected and taking the value0 otherwise). Further, from the generalized mean inequality we obtain P Ki =1 p i · i ≥ (cid:16) K m (cid:17) . In consequence, through routine calculation, we get that: E K X j =1 X j ≥ (cid:18) Km OPT + m − K − m m ( m − · (cid:18) − Km (cid:19) OPT (cid:19) = OPT2 (cid:18) Km − K m − m + K m − m (cid:19) . It remains to assess the probability that the total satisfaction obtained through Φ B isclose to its expected value. Since X j ∈ h , ( m − nK i , from Hoeffding’s inequality we get: p ǫ = Pr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K X j =1 X j − E ( K X j =1 X j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ ǫ E ( K X j =1 X j ) ≤ exp − ǫ ( E ( P Kj =1 X j )) K ( ( m − nK ) ! = exp − Kǫ ( E ( P Kj =1 X j )) (( m − n ) ! We note that since Km − K m − m ≥
0, our previous calculations show that E ( P Kj =1 X j ) ≥ OPT2 .Further, for K ≥
8, Lemma 9 (and the fact that in its proof we upper-bound OPT to be( m − n ) gives that OPT ≥ mn . Thus p ǫ ≤ exp (cid:16) − Kǫ (cid:17) . This completes the proof.25n the next theorem we will see that to have a high chance of obtaining a high qualityassignment, we need to repeat the sampling step many times. Thus, for practical purposes,by Algorithm R we mean an algorithm that repreats the sampling process a given number oftimes (this parameter is given as input) and returns the best solution found (the assignmentis created using Proposition 8).The threshold for Km , where the sampling step is (in expectation) better than the greedyalgorithm is about 0.57. Thus, by combining the two algorithms, we can guarantee anexpected approximation ratio of 0 . − ǫ , for each fixed constant ǫ . The pseudo-code ofthe combination of the two algorithms (Algorithm AR) is presented in Figure 5. Theorem 12.
For each fixed ǫ , Algorithm AR provides a (0 . − ǫ ) -approximate solutionfor the problem α B , dec - SU-Monroe with probability λ in time polynomial with respect tothe input instance size and − log(1 − λ ) .Proof. Let ǫ be a fixed constant. We are given an instance I of α B , dec - SU-Monroe . If m ≤ ǫ , we solve I using a brute-force algorithm (note that in this case the numberof alternatives is at most a fixed constant). Similarly, if H K K ≥ ǫ then we use the exactalgorithm of Betzler et al. [7] for a fixed value of K (note that in this case K is no greaterthan a certain fixed constant). We do the same if K ≤ H K K ≤ ǫ and m > ǫ then Algorithm A achieves approximation rationo worse than (1 − K m − ǫ ). We run the sampling-based algorithm −
512 log(1 − λ ) Kǫ times.The probability that a single run fails to find a solution with approximation ratio at least (1 + Km − K m − m + K m − m ) − ǫ is p ǫ ≤ exp (cid:16) − Kǫ · (cid:17) . Thus, the probability that at leastone run will find a solution with at least this approximation ratio is at least:1 − p −
512 log(1 − λ ) Kǫ ǫ = 1 − exp (cid:18) − Kǫ · · −
512 log(1 − λ ) Kǫ (cid:19) = λ. Since m ≤ ǫ , by routine calculation we see that the sampling-based algorithm withprobability λ finds a solution with approximation ratio at least (1 + Km − K m + K m ) − ǫ .By solving the equality: 12 (cid:18) Km − K m + K m (cid:19) = 1 − K m we can find the value of Km for which the two algorithms give the same approximationratio. By substituting x = Km we get equality 1 + x − x + x = 2 − x . One can calculatethat this equality has a single solution within h , i and that this solution is x ≈ . x both algorithms guarantee approximation ratio of 0 . − ǫ . For x < .
57 thedeterministic algorithm guarantees a better approximation ratio and for x > .
57, therandomized algorithm does better. 26 otation : We use the same notation as in Algorithm 2; w( · ) denotes Lambert’s W-Function. Parameters : λ ← required probability of achieving the approximation ratio equal 0 . − ǫ if H K K ≥ ǫ or K ≤ then compute the optimal solution using an algorithm of Betzler et al. [7] and return. if m ≤ ǫ then compute the optimal solution using a simple brute force algorithm and return.Φ ← solution returned by Algorithm AΦ ← run the sampling-based algorithm −
512 log(1 − λ ) Kǫ times; select the assignment of the bestqualityreturn the better assignment among Φ and Φ Figure 5: Algorithm AR—combination of Algorithms A and R.Let us now consider the case of CC. It is just as natural to try a sampling-based ap-proach for solving α B , dec - SU-CC , as we did for the Monroe variant. Indeed, as recently (andindependently) observed by Oren [35], this leads to a randomized algorithm with expectedapproximation ratio of (1 − K +1 )(1 + m ). However, since we will later see an effective, de-terministic, polynomial-time approximation scheme for α B , dec - SU-CC , there is little reasonto explore the sampling based approach.
Algorithm GM (greedy marginal improvement) was introduced by Lu and Boutilier for thecase of the Chamberlin–Courant rule. Here we generalize it to apply to Monroe’s rule aswell, and we show that it is a 1 − e approximation algorithm for α - SU-Monroe . We pointout that this approximation result for Monroe rule applies to all non-decreasing PSFs α .For the Monroe rule, the algorithm can be viewed as an extension of Algorithm B.The algorithm proceeds as follows. We start with an emtpy set S . Then we execute K iterations. In each iteration we find an alternative a that is not assigned to agents yet, andthat maximizes the value Φ S ∪{ a } α . (A certain disadvantage of this algorithm for the case ofMonroe is that it requires a large number of computations of Φ Sα ; since in Monroe’s rule eachalternative can be assigned at most nK agents in the partial assignment Φ Sα , computation ofΦ Sα is a slow process based on min-cost/max-flow algorithm.) We provide the pseudocodefor Algorithm GM in Figure 6. Theorem 13.
For any non-decreasing positional scoring function α Algorithm GM is an (1 − e ) -approximation algorithm for α - SU-Monroe .Proof.
The proof follows by applying the powerful result of Nemhauser et al. [33], which saysthat greedy algorithms achieve 1 − e approximation ratio when used to optimize nondecreas-ing submodular functions (we explain these notions formally below). The main challengein the proof is to define a function that, on one hand, satisfies the conditions of Nemhauseret al.’s result, and, on the other, models solutions for α - SU-Monroe .27 otation : Φ Sα —the partial assignement that assigns a single alternative to at most ⌈ nK ⌉ agents, that assigns to the agents only the alternatives from S , and thatmaximizes the utilitarian satisfaction ℓ α (Φ Sα ). S ← ∅ for i ← to K do a ← argmax a ∈ A \ S ℓ α (Φ S ∪{ a } α ) S ← S ∪ { a } return Φ Sα Figure 6: Pseudocode for Algorithm GM.Let A be a set of alternatives, N = [ n ] be a set of agents with preferences over A , α bean k A k -candidate DPSF, and K ≤ k A k be the number of representatives that we want toelect. We consider function z : 2 A → N defined, for each set S , S ⊆ A and k S k ≤ K , as z ( S ) = ℓ α (Φ Sα ). Clearly, z ( S ) is nondecreasing (that is, for each two sets A and B , if A ⊆ B and k B k ≤ K then z ( A ) ≤ z ( B ). Since argmax S ⊂ A, k S k = K z ( S ) is the set of winners under α -Monroe and since Algorithm GM builds the solution iteratively by greedily extendinginitially empty set S so that each iteration increases the value of z ( S ) maximally, if z weresubmodular then by the results of Nemhauser et al. [33] we would get that Algorithm GMis a (1 − e )-approximation algorithm. Thus, our goal is to show that z is submodular.Formally, our goal is to show that for each two sets S and T , S ⊂ T , and each alternative a / ∈ T it holds that z ( S ∪ { a } ) − z ( S ) ≥ z ( T ∪ { a } ) − z ( T ) (this is the formal definition ofsubmodularity). First, we introduce a notion that generalizes the notion of a partial set ofwinners S . Let s : A → N denote a function that assigns a capacity to each alternative (i.e., s gives a bound on the number of agents that a given alternative can represent). Intuitively,each set S , S ⊆ A , corresponds to the capacity function that assigns ⌈ nk ⌉ to each alternative a ∈ S and 0 to each a / ∈ S . Given a capacity function s , we define a partial solution Φ sα to be one that maximizes the total satisfaction of the agents and that satisfies the newcapacity constraints: ∀ a ∈ S k (Φ sα ) − ( a ) k ≤ s ( a ). To simplify notation, we write s ∪ { a } todenote the function such that ( s ∪ { a } )( a ) = s ( a ) + 1 and ∀ a ′ ∈ S \{ a } ( s ∪ { a } )( a ′ ) = s ( a ′ ).(Analogously, we interpret s \ { a } as subtracting one from the capacity for a ; provided itis nonzero.) Also, by s ≤ t we mean that ∀ a ∈ A s ( a ) ≤ t ( a ). We extend our function z toallow us to consider a subset of the agents only. For each subset N ′ of the agents and eachcapacity function s , we define z N ′ ( s ) to be the satisfaction of the agents in N ′ obtainedunder Φ sα . We will now prove a stronger variant of submodularity for our extended z . Thatis, we will show that for each two capacity functions s and t it holds that: s ≤ t ⇒ z N ( s ∪ { a } ) − z N ( s ) ≥ z N ( t ∪ { a } ) − z N ( t ) . (5)Our proof is by induction on N . Clearly, Equation (5) holds for N ′ = ∅ . Now, assumingthat Equation (5) holds for every N ′ ⊂ N we will prove its correctness for N . Let i denotean agent such that Φ t ∪{ a } α ( i ) = a (if there is no such agent then clearly the equation holds).28et a s = Φ sα ( i ) and a t = Φ tα ( i ). We have: z N ( t ∪ { a } ) − z N ( t ) = α (pos i ( a )) + z N \{ i } ( t ) − α (pos i ( a t )) − z N \{ i } ( t \ { a t } ) . We also have: z N ( s ∪ { a } ) − z N ( s ) ≥ α (pos i ( a )) + z N \{ i } ( s ) − α (pos i ( a s )) − z N \{ i } ( s \ { a s } ) . Since Φ tα describes an optimal representation function under the capacity restrictions t , wehave that: α (pos i ( a t )) + z N \{ i } ( t \ a t ) ≥ α (pos i ( a s )) + z N \{ i } ( t \ { a s } ) . Finally, from the inductive hypothesis for N ′ = N \ { i } we have: z N \{ i } ( s ) − z N \{ i } ( s \ { a s } ) ≥ z N \{ i } ( t ) − z N \{ i } ( t \ { a s } ) . By combining these inequalities we get: z N ( s ∪ { a } ) − z N ( s ) ≥ α (pos i ( a )) + z N \{ i } ( s ) − ( α (pos i ( a s )) + z N \{ i } ( s \ { a s } )) ≥ α (pos i ( a )) − α (pos i ( a s )) + z N \{ i } ( t ) − z N \{ i } ( t \ { a s } ) ≥ α (pos i ( a )) + z N \{ i } ( t ) − α (pos i ( a t )) − z N \{ i } ( t \ { a t } )= z N ( t ∪ { a } ) − z N ( t ) . This completes the proof.Formally speaking, Algorithm GM is never worse than Algorithm A. For Borda satis-faction function, it inherits the approximation guarantees from Algorithm A, and for othercases Theorem 13 guarantees approximation ratio 1 − e (we do not know of any guaranteesfor Algorithm A for these cases). The comparison with Algorithms B and C is not nearly aseasy. Algorithm GM is still likely better than them for satisfaction functions significantlydifferent from Borda’s, but for the Borda case our experiments show that Algorithm GMis much slower than Algorithms B and C and obtains almost the same or slightly worseresults (see Section 5). The idea of our algorithm (presented in Figure 7) is to compute a certain value x and togreedily compute an assignment that (approximately) maximizes the number of agents as-signed to one of their top- x alternatives. If after this process some agent has no alternativeassigned, we assign him or her to his or her most preferred alternative from those alreadypicked. Somewhat surprisingly, it turns out that this greedy strategy achieves high-qualityresults. (Recall that for nonnegative real numbers, Lambert’s W-function, w( x ), is definedto be the solution of the equation x = w( x ) e w( x ) .) This is very similar to the so-called MaxCover problem. Skowron and Faliszewski [42] have discussedthe connection of MaxCover to the winner determination problem under the Chamberlin–Courant votingsystem (for approval-based satisfaction functions) and provided a number of FPT approximation schemesfor it. otation : We use the same notation as in Algorithm C;num pos x ( a ) ← k{ i ∈ [ n ] \ Φ ← : pos i ( a ) ≤ x }k (the number of not-yet assignedagents that rank alternative a in one of their first x positions)Φ = {} x = ⌈ m w( K ) K ⌉ for i ← to K do a i ← argmax a ∈ A \ Φ → num pos x ( a ) foreach j ∈ [ n ] \ Φ ← doif pos j ( a i ) < x then Φ[ j ] ← a i foreach j ∈ A \ Φ ← do a ← such server from Φ → that ∀ a ′ ∈ Φ → pos j ( a ) ≤ pos j ( a ′ )Φ[ j ] ← a Figure 7: The algorithm for α B , dec - SU-CC (Algorithm P).
Lemma 14.
Algorithm P is a polynomial-time (1 − K ) K ) -approximation algorithm for α B , dec - SU-CC .Proof.
Let x = m w( K ) K . We will first give an inductive proof that, for each i , 0 ≤ i ≤ K ,after the i ’th iteration of the outer loop at most n (1 − w ( K ) K ) i agents are unassigned. Basedon this observation, we will derive the approximation ratio of our algorithm.For i = 0, the inductive hypothesis holds because n (1 − w( K ) K ) = n . For each i , let n i denote the number of unassigned agents after the i ’th iteration. Thus, after the i ’thiteration there are n i unassigned agents, each with x unassigned alternatives among his orher top- x ranked alternatives. As a result, at least one unassigned alternative is present inat least n i xm − i of top- x positions of unassigned agents. This means that after the ( i + 1)’stiteration the number of unassigned agents is: n i +1 ≤ n i − n i xm − i ≤ n i (cid:16) − xm (cid:17) = n i (cid:18) − w( K ) K (cid:19) . If for a given i the inductive hypothesis holds, that is, if n i ≤ n (cid:16) − w( K ) K (cid:17) i , then: n i +1 ≤ n (1 − w( K ) K ) i (1 − w( K ) K ) = n (cid:18) − w( K ) K (cid:19) i +1 . Thus the hypothesis holds and, as a result, we have that: n k ≤ n (cid:18) − w( K ) K (cid:19) K ≤ n (cid:18) e (cid:19) w( K ) = n w( K ) K .
Let Φ be the assignment computed by our algorithm. To compare it against the optimalsolution, it suffices to observe that the optimal solution has the value of satisfaction of at30ost OPT ≤ ( m − n , that each agent selected during the first K steps has satisfaction atleast m − x = m − m w( K ) K , and that the agents not assigned within the first K steps havesatisfaction no worse than 0. Thus it holds that: ℓ α B , dec (Φ)OPT ≥ ( n − n w( K ) K )( m − m w( K ) K )( m − n ≥ (cid:18) − w( K ) K (cid:19) (cid:18) − w( K ) K (cid:19) ≥ − K ) K .
This completes the proof.Since for each ǫ > K ǫ such that for each K > K ǫ it holds that K ) K < ǫ , and α B , dec - SU-CC problem can be solved optimally in polynomial time for eachfixed constant K (see the work of Betzler et al. [7]), there is a polynomial-time approximationscheme (PTAS) for α B , dec - SU-CC (i.e., a family of algorithms such that for each fixed r ,0 < r <
1, there is a polynomial-time r -approximation algorithm for α B , dec - SU-CC in thefamily; note that in PTASes we measure the running time by considering r to be a fixedconstant). Theorem 15.
There is a PTAS for α B , dec - SU-CC . The idea used in Algorithm P can also be used to address a generalized
SE-CC problem.We can consider the following relaxation of
SE-CC : Instead of requiring that each agent’ssatisfaction is lower-bounded by some value, we ask that the satisfactions of a significantmajority of the agents are lower-bounded by a given value. More formally, for a givenconstant δ , we introduce an additional quality metric: ℓ δ,α min (Φ) = max N ′ ⊆ N : || N ||−|| N ′|||| N || ≤ δ min i ∈ N ′ α ( pos i (Φ( i ))) . For a given 0 < δ <
1, by putting x = − m ln( δ ) K , we get (1 + ln( δ ) K )-approximation algorithmfor the ℓ δ,α min (Φ) metric.Finally, we show that Algorithm P performs very well even if the voters cast truncatedballots. Proposition 16 gives the relation between the number of positions used by thealgorithm and the approximation ratio. In Figure 8 we show this relation for some valuesof the parameters m and K . Proposition 16.
Let Q be the number of top positions in the agents’ preference ordersthat are known by the algorithm ( Q ≤ m w( K ) K ). Algorithm P that uses x = Q instead of x = ⌈ m w( K ) K ⌉ is a polynomial-time (cid:16) m − Qm − (1 − e − QKm ) (cid:17) -approximation algorithm for α B , dec - SU-CC .Proof.
Let n i denote the number of the agents not-yet-assigned until the ( i + 1)-th iterationof the algorithm. Using the same reasoning as in Lemma 14 we show that n i ≤ n (1 − Qm ) i .31 a pp r o x i m a t i o n g u a r a n t ee K =5, m =100)( K =10, m =100)( K =20, m =100)( K =30, m =100) 00.20.40.60.81 a pp r o x i m a t i o n g u a r a n t ee K =50, m =1000)( K =100, m =1000)( K =200, m =1000)( K =300, m =1000) Figure 8: The relation between the percentage of the known positions and the approximationratio of Algorithm P for α B , dec - SU-CC .As before, our proof proceeds by induction on i . It is evident that the hypothesis is correctfor i = 0. Now, assuming that n i ≤ n (1 − Qm ) i , we assess n i +1 as follows: n i +1 ≤ n i − n i Qm − i ≤ n i (cid:18) − Qm (cid:19) ≤ n (cid:18) − Qm (cid:19) i +1 .This proves the hypothesis. Thus, we can bound n K : n K ≤ n (cid:18) − Qm (cid:19) K ≤ n (cid:18) e (cid:19) QKm .This means that the satisfaction of the assignment Φ returned by our algorithm is at least: ℓ α B , dec (Φ) ≥ ( n − n K )( m − Q ) ≥ n ( m − Q )(1 − e − QKm ).In effect, it holds that: ℓ α B , dec (Φ)OPT ≥ n ( m − Q )(1 − e − QKm ) n ( m − ≥ m − Qm − (cid:16) − e − QKm (cid:17) .This completes the proof.For example, for Polish parliamentary elections ( K = 460, m = 6000), it suffices thateach voter ranks only 0 .
5% of his or her top alternatives (that is, about 30 alternatives)for the algorithm to find a solution with guaranteed satisfaction at least 90% of the one(possibly infeasible) where every voter is assigned to his or her top alternative.
To experimentally measure the quality of our approximation algorithms, we compare theresults against optimal solutions that we obtain using integer linear programs (ILPs) that32olve the Monroe and Chamberlin–Courant winner determination problem. An ILP for theMonroe rule was provided by Potthoff and Brams [36], Lu and Boutilier [25] adapted italso for the Chamberlin–Courant rule with arbitraty PSF α . For the sake of completeness,below we recall the ILP whose optimal solutions correspond to α -SU-Monroe winner sets forthe given election (we also indicate which constraints to drop to obtain an ILP for finding α -SU-CC winner sets):1. For each i , 1 ≤ i ≤ n , and each j , 1 ≤ j ≤ m we have a 0 / a ij indicatingwhether alternative a j represents agent i . For each j , 1 ≤ j ≤ m , we have a 0 / x j indicating whether alternative a j is included in the set of winners.2. Our goal is to maximize the value P ni =1 α ( pos i ( a j )) a ij subject to the following con-straints:(a) For each i and j , 1 ≤ i ≤ n, ≤ j ≤ m , 0 ≤ a ij ≤ x j (alaternative a j canrepresent agent i only if a j belongs to the set of winners)(b) For each i , 1 ≤ i ≤ n , P ≤ j ≤ m a ij = 1 (every agent is represented by exactly onealternative).(c) For each j , 1 ≤ j ≤ m , x j ⌊ nK ⌋ ≤ P ≤ i ≤ n a ij ≤ x j ⌈ nK ⌉ (each alternative eitherdoes not represent anyone or represents between ⌊ nK ⌋ and ⌈ nK ⌉ agents; if weremove these constraints then we obtain an ILP for the Chamberlin-Courantrule).(d) P nj =1 x j ≤ K (there are exactly K winners ).We used the GLPK 4.47 package (GNU Linear Programming Kit, version 4.47) to solvethese ILPs, whenever it was possible to do so in reasonable time. In this section we present the results of empirical evaluation of algorithms from Section 4.In the experiments we evaluated versions of the randomized algorithms that use exactly 100sampling steps. In all cases but one, we have used Borda PSF to measure voter satisfaction.In one case, with six candidates, we have used DPSF defined through vector (3 , , , , , C/C opt , where C is thesatisfaction obtained by a respective algorithm and C opt is the satisfaction in the optimal For the Monroe framework inequality here is equivalent to equality. We use the inequality so thatdeleting constraints from item (2c) leads to an ILP for the Chamberlin-Courant rule.
C/C ideal ,where C ideal is the satisfaction that the voters would have obtained if each of them werematched to his or her most preferred alternative. In our further experiments, where weconsidered larger elections, we were not able to compute optimal solutions, but fraction C/C ideal gives a lower bound for
C/C opt . We report this value for small elections so thatwe can see an example of the relation between
C/C opt and
C/C ideal and so that we cancompare the results for small elections with the results for the larger ones. Further, forthe case of Borda PSF the
C/C ideal fraction has a very natural interpretation: If its valuefor a given solution is v , then, on the average, in this solution each voter is matched to analternative that he or she prefers to ( m − v alternatives.In our second set of experiments, we have run our algorithms on large elections (thou-sands of agents, hundreds of alternatives), coming either from the NetFlix data set (seebelow) or generated by us using one of our models. Here we reported the average fraction C/C ideal only. We have analyzed the quality of the solutions as a function of the number ofagents, the number of candidates, and the relative number of winners (fraction
K/m ). (Thislast set of results is particularly interesting because in addition to measuring the qualityof our algorithms, it allows one to asses the size of a committee one should seek if a givenaverage satisfaction of agents is to be obtained).In the third set of experiments, we have investigated the effect of submitting trun-cated ballots (i.e., preference orders where only some of the top alternatives are ranked).Specifically, we focused on the relation between the fraction of ranked alternatives and theapproximation ratio of the algorithms. We run our experiments on relatively large instancesdescribing agents’ preferences; thus, here as in the previous set of experiments, we used Net-Flix data set and the synthetic data sets. We report the quality of the algorithms as theratio
C/C ideal .In the fourth set of experiments we have measured running times of our algorithms andof the ILP solver. Even though all our algorithms (except for the ILP based ones) arepolynomial-time, in practice some of them are too slow to be useful.
For the evaluation of the algorithms we have considered both real-life preference-aggregationdata and synthetic data, generated according to a number of election models. The experit-ments reported in this paper predate the work of Mattei and Walsh [29] on gathering a largecollection of data sets with preference data, but we mention that the conference version ofthis paper contributed several data sets to their collection.
We have used real-life data regarding people’s preference on sushi types, movies, collegecourses, and competitors’ performance in figure-skating competitions. One of the majorproblems regarding real-life preference data is that either people express preferences over avery limited set of alternatives, or their preference orders are partial. To address the latter34ssue, for each such data set we complemented the partial orders to be total orders usingthe technique of Kamishima [23]. (The idea is to complete each preference order based onthose reported preference orders that appear to be similar.)Some of our data sets contain a single profile, whereas the others contain multipleprofiles. When preparing data for a given number m of candidates and a given number n of voters from a given data set, we used the following method: We first uniformly atrandom chose a profile within the data set, and then we randomly selected n voters and m candidates. We used preference orders of these n voters restricted to these m candidates. Sushi Preferneces.
We used the set of preferences regarding sushi types collected byKamishima [23]. Kamishima has collected two sets of preferences, which we call S1 and S2 . Data set S1 contains complete rankings of 10 alternatives collected from 5000 voters.S2 contains partial rankings provided by 5000 voters over a set of 100 alternatives (eachvote ranks 10 alternatives). We used Kamishima [23] technique to obtain total rankings. Movie Preferences.
Following Mattei et al. [28], we have used the NetFlix data set ofmovie preferences (we call it Mv ). NetFlix data set contains ratings collected from about480 thousand distinct users regarding 18 thousand movies. The users rated movies by givingthem a score between 1 (bad) and 5 (good). The set contains about 100 million ratings. Wehave generated 50 profiles using the following method: For each profile we have randomlyselected 300 movies, picked 10000 users that ranked the highest number of the selectedmovies, and for each user we have extended his or her ratings to a complete preferenceorder using the method of Kamishima [23]. Course Preferences.
Each year the students at the AGH University choose courses thatthey would like to attend. The students are offered a choice of six courses of which theyhave to attend three. Thus the students are asked to give an unordered set of their threetop-preferred courses and a ranking of the remaining ones (in case too many students selecta course, those with the highest GPA are enrolled and the remaining ones are moved to theirless-preferred courses). In this data set, which we call Cr , we have 120 voters (students)and 6 alternatives (courses). However, due to the nature of the data, instead of using Bordacount PSF as the satisfaction measure, we have used the vector (3 , , , , , Figure Skating.
This data set, which we call Sk , contains preferences of the judgesover the performances in a figure-skating competitions. The data set contains 48 profiles,each describing a single competition. Each profile contains preference orders of 9 judges overabout 20 participants. The competitions include European skating championships, OlympicGames, World Junior, and World Championships, all from 1998 . (Note that while in figureskating judges provide numerical scores, this data set is preprocessed to contain preferenceorders.) The sushi data set is available under the following url: This data set is available under the following url: http://rangevoting.org/SkateData1998.txt . .1.2 Synthetic Data For our tests, we have also used profiles generated using three well-known distributions ofpreference orders.
Impartial Culture.
Under the impartial culture model of preferences (which we denote IC ), for a given set A of alternatives, each voter’s preference order is drawn uniformly atrandom from the set of all possible total orders over A . While not very realistic, profiles gen-erated using impartial culture model are a standard testbed of election-related algorithms. Polya-Eggenberger Urn Model.
Following McCabe-Dansted and Slinko [31] andWalsh [44], we have used the Polya-Eggenberger urn model [4] (which we denote Ur ).In this model we generate votes as follows. We have a set A of m alternatives and an urnthat initially contains all m ! preference orders over A . To generate a vote, we simply ran-domly pick one from the urn (this is our generated vote), and then—to simulate correlationbetween voters—we return a copies of this vote to the urn. When generating an electionwith m candidates using the urn model, we have set the parameter a so that am ! = 0 . b ; we mentionthat those authors use much higher values of b but we felt that too high a value of b leadsto a much too strong correlation between votes). Generalized Mallow’s Model.
We refer to this data set as Ml . Let ≻ and ≻ ′ be twopreference orders over some alternative set A . Kendal-Tau distance between ≻ and ≻ ′ ,denoted d K ( ≻ , ≻ ′ ), is defined as the number of pairs of candidates x, y ∈ A such that either x ≻ y ∧ y ≻ ′ x or y ≻ x ∧ x ≻ ′ y .Under Mallow’s distribution of preferences [27] we are given two parameters: A center preference order ≻ and a number φ between 0 and 1. The model says that the probability ofgenerating preference order ≻ ′ is proportional to the value φ d K ( ≻ , ≻ ′ ) . To generate preferenceorders following Mallow’s distribution, we use the algorithm given by Lu and Boutilier [26].In our experiments, we have used a mixture of Mallow’s models. Let A be a set ofalternatives and let ℓ be a positive integer. This mixture model is parametrized by threevectors, Λ = ( λ , . . . , λ ℓ ) (where each λ i is between 0 and 1, and P ℓi =1 λ = 1), Φ =( φ , . . . , φ ℓ ) (where each φ i is a number between 0 and 1), and Π = ( ≻ , . . . , ≻ ℓ ) (whereeach ≻ i is a preference order over A ). To generate a vote, we pick a random integer i ,1 ≤ i ≤ ℓ (each i is chosen with probability λ i ), and then generate the vote using Mallow’smodel with parameters ( ≻ i , φ i ).For our experiments, we have used a = 5, and we have generated vectors Λ, Φ, and Πuniformly at random. We now present the results of our experiments on small elections. For each data set, wegenerated elections with the number of agents n = 100 ( n = 9 for data set Sk because thereare only 9 voters there) and with the number of alternatives m = 10 ( m = 6 for data set Cr S1 .
94 0 . ≈ . .
99 0 .
99 1 . ≈ . .
99 0 . S2 .
95 0 .
99 1 . ≈ . .
99 1 . ≈ . .
98 0 . Mv . ≈ . . ≈ . .
98 1 . ≈ . . ≈ . Cr .
98 0 .
99 1 . ≈ . .
99 1 . ≈ . . ≈ . Sk . ≈ . . ≈ . .
94 1 . ≈ . .
85 0 . IC .
94 0 . ≈ . .
99 0 .
99 1 . ≈ . .
99 0 . Ml .
94 0 .
99 1 . .
99 0 .
99 1 . ≈ . .
99 0 . Ur .
95 0 . ≈ . .
99 0 .
99 1 . .
99 0 .
97 0 . C/C opt )for the small instances of data and for K = 3 ( K = 2 for Cr ); m = 10 ( m = 6 for Cr ); n = 100 ( n = 9 for Sk ). Monroe CCA B C GM R C GM P R S1 . ≈ . . .
99 0 .
99 1 . ≈ . .
97 0 . S2 .
94 0 . ≈ . .
99 0 .
99 1 . ≈ . . ≈ . Mv .
95 0 .
99 1 . ≈ . .
98 1 . ≈ . . ≈ . Cr . ≈ . . ≈ . .
99 1 . . . . Sk . ≈ . . ≈ . .
88 1 . . . ≈ . IC .
95 0 . ≈ . .
99 0 .
99 1 . ≈ . .
99 0 . Ml .
95 0 . ≈ . .
99 0 .
99 1 . ≈ . .
98 0 . Ur .
96 0 . ≈ . . ≈ . . ≈ . .
96 0 . C/C opt )for the small instances of data and for K = 6 ( K = 4 for Cr ); m = 10 ( m = 6 for Cr ); n = 100 ( n = 9 for Sk ).because there are only 6 alternatives there) using the method described in Section 5.1.1 forthe real-life data sets, and in the natural obvious way for synthetic data. For each algorithmand for each data set we ran 500 experiments on different instances for K = 3 (for the Cr data set we used K = 2) and 500 experiments for K = 6 (for Cr we set K = 4). ForAlgorithm C (both for Monroe and for CC) we set the parameter d , describing the numberof assignment functions computed in parallel, to 15. The results (average fractions C/C opt and
C/C ideal ) for K = 3 are given in Tables 1 and 3; the results for K = 6 are given inTables 2 and 4 (they are almost identical as for K = 3). For each experiment in this sectionwe also computed the standard deviation; it was always on the order of 0 .
01. The resultslead to the following conclusions: 37onroe CCA B C GM R C GM P R S1 .
85 0 .
89 0 . .
89 0 .
89 0 .
92 0 .
89 0 .
91 0 . S2 .
85 0 .
89 0 .
89 0 .
89 0 .
89 0 .
93 0 . .
91 0 . Mv .
88 0 .
92 0 .
92 0 .
92 0 .
91 0 .
97 0 .
92 0 .
93 0 . Cr .
94 0 .
97 0 .
96 0 .
96 0 .
96 0 .
97 0 .
97 0 .
97 0 . Sk .
96 0 .
96 0 .
97 0 .
97 0 .
91 1 . .
97 0 .
82 0 . IC . .
84 0 .
85 0 .
84 0 .
84 0 .
85 0 .
83 0 .
84 0 . Ml .
83 0 .
88 0 .
88 0 . .
88 0 .
92 0 .
90 0 .
89 0 . Ur . .
85 0 .
86 0 .
87 0 .
85 0 . .
87 0 .
87 0 . C/C ideal ) for the small instances of data and for K = 3 ( K = 2 for Cr ); m = 10 ( m = 6for Cr ); n = 100 ( n = 9 for Sk ).Monroe CCA B C GM R C GM P RS1 0 .
91 0 .
96 0 .
96 0 .
95 0 .
95 0 .
98 0 .
98 0 .
96 0 . .
88 0 .
93 0 .
93 0 .
93 0 .
93 0 .
98 0 .
98 0 .
96 0 . .
85 0 .
89 0 .
89 0 .
89 0 .
88 0 .
99 0 .
99 0 .
97 0 . .
95 0 .
98 0 .
99 0 .
99 0 .
98 1 . . . . .
91 0 .
92 0 .
92 0 .
92 0 .
81 1 . . . ≈ . .
91 0 .
95 0 .
95 0 .
94 0 .
95 0 .
96 0 .
96 0 .
95 0 . .
89 0 .
94 0 .
94 0 .
94 0 .
93 0 .
97 0 .
98 0 .
95 0 . .
91 0 .
95 0 .
95 0 .
94 0 .
95 0 .
98 0 .
98 0 .
94 0 . C/C ideal ) for the small instances of data and for K = 6 ( K = 4 for Cr ); m = 10 ( m = 6for Cr ); n = 100 ( n = 9 for Sk ).1. For the case of Monroe, already Algorithm A obtains very good results, but nonethe-less Algorithms B and C improve noticeably upon Algorithm A. In particular, Algo-rithm C (for d = 15) obtains the highest satisfaction on all data sets and in almostall cases was able to find an optimal solution.2. Both for Monroe and for CC, Algorithm R gives slightly worse solutions than Algo-rithm C.3. The results do not seem to depend on the data sets used in the experiments (the onlyexception is Algorithm R for the Monroe system on data set Sk ; however Sk has only9 voters so it can be viewed as a border case).38 .3 Evaluation on Larger Instances . . . . . . qu a lit yo f t h ea l go r it h m ( C / C i d e a l )
50 100 150 200 250 300 number of alternatives m Algorithm C for U R Algorithm A for U R Algorithm C for M V Algorithm R for U R Algorithm A for M V Algorithm R for M V . . . . qu a lit yo f t h ea l go r it h m ( C / C i d e a l )
50 100 150 200 250 300 number of alternatives m Algorithm C for M L Algorithm C for ICAlgorithm A for M L Algorithm R for M L Algorithm R for ICAlgorithm A for IC
Figure 9: The relation between the number of alternatives m and the quality of the algo-rithms C/C ideal for the Monroe system;
K/m = 0 . n = 1000. q u a li t y o f t h e a l g o r i t h m ( C / C i d e a l )
50 100 150 200 250 300number of alternatives m Algorithm C for M V Algorithm C for U R Algorithm R for M V Algorithm R for U R q u a li t y o f t h e a l g o r i t h m ( C / C i d e a l )
50 100 150 200 250 300number of alternatives m Algorithm R for M L Algorithm C for M L Algorithm C for ICAlgorithm R for IC
Figure 10: The relation between the number of alternatives m and the quality of thealgorithms C/C ideal for the Chamberlin–Courant system;
K/m = 0 . n = 1000.For experiments on larger instances we needed data sets with at least n = 10000 agents.Thus we used the NetFlix data set and synthetic data. (Additionally, we run the subsetof experiments (for n ≤ S2 data set.) For the Monroe rule we presentresults for Algorithm A, Algorithm C, and Algorithm R, and for the Chamberlin–Courantrule we present results for Algorithm C and Algorithm R. We limit the set of algorithmsfor the sake of the clarity of the presentation. For Monroe we chose Algorithm A becauseit is the simplest and the fastest one, Algorithm C because it is the best generalization ofAlgorithm A that we were able to run in reasonable time, and Algorithm R to compare arandomized algorithm to deterministic ones. For the Chamberlin–Courant rule we choseAlgorithm C because it is, intuitively, the best one, and we chose Algorithm R for the samereason as in the case of Monroe.First, for each data set and for each algorithm we fixed the value of m and K andfor each n ranging from 1000 to 10000 with the step of 1000 we run 50 experiments. Werepeated this procedure for 4 different combinations of m and K : ( m = 10, K = 3),39 . . . . . . qu a lit yo f t h ea l go r it h m ( C / C i d e a l )
10 20 30 40 50 ratio
K/m [%]Algorithm C for U R Algorithm A for U R Algorithm C for M V Algorithm R for U R Algorithm A for M V Algorithm R for M V . . . . qu a lit yo f t h ea l go r it h m ( C / C i d e a l )
10 20 30 40 50 ratio
K/m [%]Algorithm C for M L Algorithm C for ICAlgorithm R for ICAlgorithm A for M L Algorithm R for M L Algorithm A for IC
Figure 11: The relation between the ratio
K/m and the quality of the algorithms
C/C ideal for the Monroe system; m = 100; n = 1000. q u a li t y o f t h e a l g o r i t h m ( C / C i d e a l )
10 20 30 40 50ratio K / m [%]Algorithm R for M V Algorithm C for M V Algorithm R for U R Algorithm C for U R q u a li t y o f t h e a l g o r i t h m ( C / C i d e a l )
10 20 30 40 50ratio K / m [%]Algorithm R for M L Algorithm C for M L Algorithm C for ICAlgorithm R for IC
Figure 12: The relation between the ratio
K/m and the quality of the algorithms
C/C ideal for the Chamberlin–Courant system; m = 100; n = 1000.( m = 10, K = 6), ( m = 100, K = 30) and ( m = 100, K = 60). We measured the statisticalcorrelation between the number of voters and the quality of the algorithms C/C ideal . TheANOVA test in most cases showed that there is no such correlation. The only exceptionwas S2 data set, for which we obtained an almost negligible correlation. For example, for( m = 10 , K = 3) Algorithm C under data set S2 for Monroe’s system for n = 5000 gave C/C ideal = 0 .
88, while for n = 100 (in the previous section) we got C/C ideal = 0 .
89. Thuswe conclude that in practice the number of agents has almost no influence on the qualityof the results provided by our algorithms.Next, we fixed the number of voters n = 1000 and the ratio K/m = 0 .
3, and for each m ranging from 30 to 300 with the step of 30 (naturally, as m changed, so did K to maintainthe ratio K/m ), we run 50 experiments. We repeated this procedure for
K/m = 0 .
6. Therelation between m and C/C ideal for Mv and Ur , under both the Monroe rule and theChamberlin–Courant rule, is given in Figures 9 and 10 (the results for K/m = 0 . n = 1000 and m = 100, and for each K/m ranging from 0 . . . K/m and thequality
C/C ideal is presented in Figures 11 and 12.For the case of Chamberlin–Courant system, increasing the size of the committee tobe elected improves overall agents’ satisfaction. Indeed, since there are no constraintson the number of agents matched to a given alternative, a larger committee means moreopportunities to satisfy the agents. For the Monroe rule, a larger committee may lead to alower total satisfaction. This happens if many agents like a particular alternative a lot, butonly some of them can be matched to this alternative and others have to be matched totheir less preferred ones. Nonetheless, we see that Algorithm C achieves
C/C ideal = 0 . K/m = 0 . The purpose of our third set of experiments was to see how our algorithm behave in practicalsettings with truncated ballotrs. We conducted this part of evaluation on relatively largeinstances, including n = 1000 agents and up to m = 100 alternatives. Thus, in this setof experiments, we used the same sets of data as in the previous subsection: the Netflixdata set and the synthetic distributions. Similarly, we evaluated the same algorithms:Algorithm A, C, and R for the case of Monroe’s system, and Algorithm C, and R for thecase of the Chamberlin–Courant system.For each data set and for each algorithm we run experiments for 3 independent settingswith different values of the parameters describing the elections: (1) m = 100, K = 20, (2) m = 100, K = 10, and (3) m = 20, K = 4. For each setting we run the experiments for thevalues of P (the number of known positions) varying between 1 and m .For each algorithm, data set, setting and each value of P we run 50 independent exper-iments in the following way. From a data set we sampled a sub-profile of the appropriatesize n × m . We truncated this profile to the P first positions. We run the algorithm for thetruncated profile and calculated the quality ratio C/C ideal . When calculating
C/C ideal weassumed the worst case scenario, i.e., that the satisfaction of the agent from an alternativeoutside of his/her first P positions is equal to 0. In other words, we used the positionalscoring function described by the following vector: h m − , m − , . . . , m − P, , . . . i . Next,we averaged the values of C/C ideal over all 50 experiments.The relation between the percentage of the known positions in the preference profileand the average quality of the algorithm for the Monroe and Chamberlin–Courant systemsare plotted in Figures 13 and 14, respectively. We omit the plots for Mallow’s model, as inthis case we obtained almost identical results as for the Urn model. We have the followingconclusions. 41 . . . . q u a li t y o f t h e a l g . ( C / C i d e a l ) Mv ( K =20, m =100)Algorithm C for Mv ( K =10, m =100)Algorithm C for Mv ( K =4, m =20) 00 . . . . q u a li t y o f t h e a l g . ( C / C i d e a l ) Ur ( K =20, m =100)Algorithm C for Ur ( K =10, m =100)Algorithm C for Ur ( K =4, m =20) 00 . . . . q u a li t y o f t h e a l g . ( C / C i d e a l ) IC ( K =20, m =100)Algorithm C for IC ( K =10, m =100)Algorithm C for IC ( K =4, m =20)00 . . . . q u a li t y o f t h e a l g . ( C / C i d e a l ) Mv ( K =20, m =100)Algorithm A for Mv ( K =10, m =100)Algorithm A for Mv ( K =4, m =20) 00 . . . . q u a li t y o f t h e a l g . ( C / C i d e a l ) Ur ( K =20, m =100)Algorithm A for Ur ( K =10, m =100)Algorithm A for Ur ( K =4, m =20) 00 . . . . q u a li t y o f t h e a l g . ( C / C i d e a l ) IC ( K =20, m =100)Algorithm A for IC ( K =10, m =100)Algorithm A for IC ( K =4, m =20)00 . . . . q u a li t y o f t h e a l g . ( C / C i d e a l ) Mv ( K =20, m =100)Algorithm R for Mv ( K =10, m =100)Algorithm R for Mv ( K =4, m =20) 00 . . . . q u a li t y o f t h e a l g . ( C / C i d e a l ) Ur ( K =20, m =100)Algorithm R for Ur ( K =10, m =100)Algorithm R for Ur ( K =4, m =20) 00 . . . . q u a li t y o f t h e a l g . ( C / C i d e a l ) IC ( K =20, m =100)Algorithm R for IC ( K =10, m =100)Algorithm R for IC ( K =4, m =20) Figure 13: The relation between the percentage of known positions
P/m [%] and the qualityof the algorithm
C/C ideal for Algorithms C, A, and R for Monroe’s system. Each row ofthe plots describes one algorithm; each column describes one data set; n = 1000. (Resultsfor the Mallows model are similar to those for the urn model and are omitted for clarity.)1. All the algorithms require only small number of the top positions to achieve their bestquality. Here, the deterministic algorithms are superior.2. The small elections with synthetic distributions appear to be the worst case scenario—in such case we require the knowledge of about 40% of the top positions to obtainthe highest approximation ratios of the algorithms. In the case of the NetFlix dataset, even on small instances the deterministic algorithms require only about 8% of thetop positions to get their best quality (however the quality is already high for 3-5% ofthe top positions). For the larger number of the alternatives, the algorithms do notrequire more than 3% of the top positions to reach their top results.3. Algorithm C does not only give the best quality but it is also most immune to thelack of knowledge. These results are more evident for the case of the Monroe system.42 . . . . qu a lit yo f t h ea l g . ( C / C i d e a l ) % of known positionsAlgorithm C for M V ( K =20, m =100)Algorithm C for M V ( K =10, m =100)Algorithm C for M V ( K =4, m =20) . . . . qu a lit yo f t h ea l g . ( C / C i d e a l ) % of known positionsAlgorithm C for U R ( K =20, m =100)Algorithm C for U R ( K =10, m =100)Algorithm C for U R ( K =4, m =20) . . . . qu a lit yo f t h ea l g . ( C / C i d e a l ) % of known positionsAlgorithm C for IC ( K =20, m =100)Algorithm C for IC ( K =10, m =100)Algorithm C for IC ( K =4, m =20) . . . . qu a lit yo f t h ea l g . ( C / C i d e a l ) % of known positionsAlgorithm R for M V ( K =20, m =100)Algorithm R for M V ( K =10, m =100)Algorithm R for M V ( K =4, m =20) . . . . qu a lit yo f t h ea l g . ( C / C i d e a l ) % of known positionsAlgorithm R for U R ( K =20, m =100)Algorithm R for U R ( K =10, m =100)Algorithm R for U R ( K =4, m =20) . . . . qu a lit yo f t h ea l g . ( C / C i d e a l ) % of known positionsAlgorithm R for IC ( K =20, m =100)Algorithm R for IC ( K =10, m =100)Algorithm R for IC ( K =4, m =20) Figure 14: The relation between the percentage of known positions
P/m [%] and the qualityof the algorithm
C/C ideal for Algorithms C and R for the Chamberlin–Courant system. Eachrow of the plots describes one algorithm; each column describes one data set; n = 1000.(Results for the Mallows model are similar to those for the urn model and are omitted forclarity.) In our final set of experiments, we have measured running times of our algorithms on thedata set Mv . We have used a machine with Intel Pentium Dual T2310 1.46GHz processor r unn i n g t i m e o f t h e a l go r i t h m [ s ec o nd s ]
50 100 150 200 250 300 350 400number of agents n ILP for CC ( k = 3 , m = 10)ILP for Monroe ( k = 3 , m = 10)ILP for CC ( k = 9 , m = 30)ILP for Monroe ( k = 9 , m = 30) Figure 15: The running time of the standard ILP solver for the Monroe and for theChamberlin–Courant systems. For Monroe’s system, for K = 9 , m = 30, and for n ≥ = 10, K = 3 m = 10, K = 6 n = 2000 6000 10000 2000 6000 10000 M o n r o e A 0 .
01 0 .
03 0 .
05 0 .
01 0 .
04 0 .
07B 0 .
08 0 . . . . .
6C 1 . . . . . . . CC C 0 .
02 0 .
07 0 .
12 0 .
05 0 .
14 0 . .
003 0 .
009 0 .
015 0 .
003 0 .
01 0 . .
009 0 .
032 0 .
05 0 .
008 0 .
02 0 .
05R 0 .
014 0 .
04 0 .
065 0 .
02 0 .
06 0 . m = 100, K = 30 m = 100, K = 60 n = 2000 6000 10000 2000 6000 10000 M o n r o e A 0 . . . . . .
9B 0 . . . CC C 4 . . .
06 0 . . .
09 0 . .
7P 0 .
03 0 . .
26 0 .
03 0 . .
2R 0 .
06 0 .
24 0 .
45 0 . . . K = 9 , m = 30 , n = 100 some of the experiments timed out after1 hour, and for K = 9 , m = 30 , n = 200 none of the experiments finished within one day.Thus we conclude that the real application of the ILP-based algorithm is very limited.Example running times of the other algorithms for some combinations of n , m , and K are presented in Table 5. For the case of CC, essentially all the algorithms are veryfast and the quality of computed solutions is the main criterion in choosing among them.For the case of Monroe, the situation is more complicated. While for small elections allthe algorithms are practical, for elections with thousands of voters, using Algorithm GMbecomes problematic. Indeed, even Algorithm C can be seen as a bit too slow if one expectsimmediate results. On the other hand, Algorithms A and B seem perfectly practical and,as we have seen in the previous experiments, give high-quality results.44 onroe Chamberlin and Courant General modeldissat. satisfaction dissat. satisfaction dissat. satisfaction u t il. Inapprox.Theorem 1 Goodapprox. Inapprox.Theorem 3 Goodapprox. Inapprox.Theorem 1Theorem 3 Open problem e ga l. Inapprox.Theorem 2 Inapprox.Theorem 5 Inapprox.Theorem 4 Inapprox.Theorem 7 Inapprox.Theorem 2Theorem 4 Inapprox.Theorem 5Theorem 7
Table 6: Summary of approximability results for the Monroe and Chamberlin-Courantmultiwinner voting systems and for the general resource allocation problem.
We have defined a certain resource allocation problem and have shown that it generalizesthe problem of finding winners for the multiwinner voting rules of Monroe and of Cham-berlin and Courant. Since it is known that the winners for these voting rules are hard tocompute [7,25,37,42,43], we focused on finding approximate solutions. We have shown thatif we try to optimize agents’ dissatisfaction, then our problems are hard to approximate upto any constant factor. The same holds for the case where we focus on the satisfaction ofthe least satisfied agent. However, for the case of optimizing total satisfaction, we suggestgood approximation algorithms. In particular, for the Monroe system we suggest a random-ized algorithm that for the Borda score achieves an approximation ratio arbitrarily close to0 .
715 (and much better in many real-life settings), and (1 − e )-approximation algorithm forarbitrary positional scoring function. For the Chamberlin-Courant system, we have showna polynomial-time approximation scheme (PTAS).In Table 6 we present the summary of our (in)approximability results. In Table 7we present specific results regarding our approximation algorithms for the utilitariansatisfaction-based framework. In particular, the table clearly shows that for the case ofMonroe, Algorithms B and C are not much slower than Algorithm A but offer a chance ofimproved peformance. Algorithm GM is intuitively even more appealing, but achieves thisat the cost of high time complexity. For the case of Chamberlin-Courant rule, theoreticalresults suggest using Algorithm P (however, see below).We have provided experimental evaluation of the algorithms for computing the winnersets both for the Monroe and Chamberlin–Courant rules . While finding solutions for theserules is computationally hard in the worst case, it turned out that in practice we can obtainvery high quality solutions using very simple algorithms. Indeed, both for the Monroe andChamberlin-Courant rules we recommend using Algorithm C (or Algorithm A on very largeMonroe elections). Our experimental evaluation confirms that the algorithms work verywell in case of truncated ballots. We believe that our results mean that (approximationsof) the Monroe and Chamberlin–Courant rules can be used in practice.Our work leads to a number of further research directions. First, it would be veryinteresting to find a better upper bound on the quality of solutions for the (satisfaction-based) Monroe and Chamberlin–Courant systems (with Borda PSF) than the simple n ( m − lgorithm Approximation Runtime Reference M o n r o e A 1 − K − m − − H K K Kmn
Lemma 9B as in Algorithm A
Kmn + O (Φ S ) Lemma 9C as in Algorithm A dKmn + dO (Φ S ) Lemma 9GM as in Alg. A for Borda PSF;1 − e for others KmO (Φ S ) Theorem 13R (1 + Km − K m − K m − m ) | log(1 − λ ) | Kǫ O (Φ S ) Lemma 11AR 0 .
715 max(A , R) Theorem 12 CC PTAS Theorem 15P 1 − K ) K nm w( K ) Lemma 14GM 1 − e Kmn
Lu and Boutilier [25]C as in Algorithm GM dKm ( n +log dm ) Lu and Boutilier [25]R (1 − K +1 )(1 + m ) | log(1 − λ ) | ǫ n Oren [35]
Table 7: A summary of the algorithms studied in this paper. The top of the table regardsalgorithms for Monroe’s rule and the bottom for the Chamberlin–Courant rule. In column“Approximation” we give currently known approximation ratio for the algorithm underBorda PSF, on profiles with m candidates and where the goal is to select a committee ofsize K . Here, O (Φ S ) = O ( n ( K +log n )) is the complexity of finding a partial representationfunction with the algorithm of Betzler et al. [7]. w( · ) denotes Lambert’s W-Function.1) bound that we use (where n is the number of voters and m is the number of candidates).We use a different approach in our randomized algorithm, but it would be much moreinteresting to find a deterministic algorithm that beats the approximation ratios of ouralgorithms. One of the ways of seeking such a bound would be to consider Monroe’s rulewith “exponential” Borda PSF, that is, with PSF of the form, e.g., (2 m − , m − , . . . , t -approval PSF’s α t , which (in the satisfaction-based variant) are defined as follows: α t ( i ) = 1if i ≤ t and otherwise α t ( i ) = 0. Results for this case for the Chamberlin–Courant rule arepresented in the paper of Skowron and Faliszewski [42].On a more practical side, it would be interesting to develop our study of truncatedballots. Our results show that we can obtain very high approximation ratios even whenvoters rank only relatively few of their top candidates. For example, to achieve 90% ap-proximation ratio for the satisfaction-based Monroe system in Polish parliamentary election( K = 460 , m = 6000), each voter should rank about 8 .
7% of his or her most-preferred can-didates. However, this is still over 500 candidates. It is unrealistic to expect that the voterswould be willing to rank this many candidates. Thus, how should one organize Monroe-based elections in practice, to balance the amount of effort required from the voters andthe quality of the results? 46inally, going back to our general resource allocation problem, we note that we do nothave any positive results for it (the negative results, of course, carry over from the morerestrictive settings). Is it possible to obtain some good approximation algorithm for theresource allocation problem (in the utilitarian satisfaction-based setting) in full generality?
Acknowledgements
This paper is based on two extended abstracts, presented at IJCAI-2013 and AAMAS-2012. We would like to thank the reviewers from these two conferencesfor very useful feedback. The authors were supported in part by Polands National ScienceCenter grants UMO-2012/06/M/ST1/00358, DEC-2011/03/B/ST6/01393, and by AGHUniversity of Science and Technology grant 11.11.120.865. Piotr Skowron was also supportedby EUs Human Capital Program ”National PhD Programme in Mathematical Sciences”carried out at the University of Warsaw.
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