aa r X i v : . [ c s . G T ] J a n Equitable Division of a Path
Neeldhara Misra , Chinmay Sonar , P. R. Vaidyanathan , and Rohit Vaish Indian Institute of Technology Gandhinagar, India, [email protected] University of California, Santa Barbara, USA, [email protected] Technische Universität Wien, Austria, [email protected] Tata Institute of Fundamental Research, [email protected]
Abstract
We study fair resource allocation under a connectedness constraint wherein a set of indivisibleitems are arranged on a path and only connected subsets of items may be allocated to the agents. Anallocation is deemed fair if it satisfies equitability up to one good (EQ1), which requires that agents’utilities are approximately equal. We show that achieving EQ1 in conjunction with well-studiedmeasures of economic efficiency (such as Pareto optimality, non-wastefulness, maximum egalitarianor utilitarian welfare) is computationally hard even for binary additive valuations. On the algorith-mic side, we show that by relaxing the efficiency requirement, a connected EQ1 allocation can becomputed in polynomial time for any given ordering of agents, even for general monotone valua-tions. Interestingly, the allocation computed by our algorithm has the highest egalitarian welfareamong all allocations consistent with the given ordering. On the other hand, if efficiency is required,then tractability can still be achieved for binary additive valuations with interval structure . On ourway, we strengthen some of the existing results in the literature for other fairness notions such asenvy-freeness up to one good (EF1), and also provide novel results for negatively-valued items or chores . The question of how to fairly divide a set of resources among agents has been extensively studiedin economics, mathematics, and computer science. The formal treatment of such resource allocationproblems—commonly referred to as fair division —dates back several decades (Steinhaus, 1948). Thereis now a rich literature on fair division problems (Brams and Taylor, 1996; Moulin, 2004; Brandt et al.,2016), comprising of a variety of solution concepts and associated existential and computational results.Many of these insights have found impressive practical applications such as in rent division (Su, 1999),credit assignment (De Clippel et al., 2008), and cluster computing (Ghodsi et al., 2011).Many real-world resource allocation problems exhibit a natural spatial or temporal structure, andin such scenarios, it is desirable to have contiguous allocations. For example, when allocating super-computing time, a contiguous processing window is preferable over one that involves multiple restarts.Similarly, when assigning office space in a department building, each research group might prefer acontiguous segment of rooms for ease of communication.In this work, we study the seemingly conflicting goals of fairness and contiguity in the contextof allocating indivisible resources (or goods). Specifically, we consider a set of indivisible goods thatare represented by the vertices of a path graph, and require that each agent is allocated a connectedsubgraph. Fair allocation of indivisible goods has received growing interest within both artificial in-telligence as well as theoretical computer science literature (Aziz et al., 2015; Bouveret and Lemaître,2016; Cole and Gkatzelis, 2018; Kurokawa et al., 2018; Caragiannis et al., 2019), motivated, in part,by notable real-world applications such as course allocation (Budish, 2011) and property division1Pruhs and Woeginger, 2012). The research area has been further popularized by the website
Splid-dit ( ) that provides implementations of provably fair algorithms for a wide arrayof resource allocation problems (Goldman and Procaccia, 2015).While there are countless formulations of what it means to be fair, each with its own merit, inthis work we focus on one well-established notion of fairness called equitability (Dubins and Spanier,1961). An equitable allocation is one in which agents derive equal utilities from their assigned shares.Equitability is a particularly compelling fairness criterion in settings such as dividing climate changeresponsibilities among countries (Traxler, 2002) and in designing taxation policies. It also enjoys em-pirical support, as lab experiments and an online user study have found that equitability—or “aver-sion of interpersonal inequity”’—can be an important predictor of the perceived fairness of an alloca-tion, possibly more so than the classic “intrapersonal” criterion of envy-freeness (Herreiner and Puppe,2009; Gal et al., 2017). Equitability is also a key property in the well-known adjusted winner algo-rithm (Brams and Taylor, 1996) which has been applied to divorce settlements.For indivisible items, perfect equitability may not be possible, which motivates the need for a naturalrelaxation called equitability up to one good (EQ1) (Freeman et al., 2019). This notion requires that theinequity between any pair of agents can be eliminated by removing some item from the happier agent’sbundle. Since an empty allocation is vacuously fair, the study of fairness notions is often coupled with economic efficiency . To this end, we study EQ1 alongside various efficiency measures such as Paretooptimality, non-wastefulness, and maximum egalitarian (max-min) or utilitarian (sum) welfare (seePreliminaries for the relevant definitions).The study of connected fair allocations of general graphs was initiated by Bouveret et al. (2017)with a focus on other fairness notions such as envy-freeness, proportionality, and maximin share. Con-currently, Suksompong (2019) showed that for a path graph, a connected and approximately equitableallocation always exists and can be efficiently computed. This work also provided a non-constructiveproof of existence of egalitarian-optimal and approximately equitable allocations, but did not considerother efficiency notions. Importantly, the notion of approximate equitability in Suksompong’s work isstrictly weaker than EQ1, and as we observe later, his algorithm could fail to find EQ1 allocations evenwhen such allocations are known to exist. Thus, the existential and computational questions pertainingto EQ1 allocations remain unanswered by prior work. Our Contributions:
We initiate the study of EQ1 allocations under connectedness constraints andmake the following contributions:1.
Hardness results for EQ1 and efficient allocations : We show that checking the existence ofa connected EQ1 allocation satisfying any of the aforementioned efficiency measures is NP-hardeven under binary additive valuations (Theorems 1 and 2 and Corollary 1). All of our resultsfollow from a single construction that also has implications for other fairness notions such as envy-freeness up to one good (EF1) as well as negatively-valued items (or chores ).2.
Algorithmic result for complete EQ1 allocations : By relaxing the efficiency condition andonly requiring completeness (i.e., not leaving any good unassigned), we obtain a polynomial-timealgorithm for computing a connected EQ1 allocation whose egalitarian welfare is the highestamong all allocations that are consistent with a given ordering of agents (Theorem 3). This re-solves an open problem of Suksompong (2019). Notably, our algorithm applies to any instancewith monotone (possibly non-additive) valuations.3.
Structured preferences : We provide an efficient algorithm for checking the existence of a con-nected, non-wasteful, and EQ1 allocation when agents have binary additive valuations with ex-tremal interval structure (Theorem 5). 2
Related Work
Fair division problems have been classically studied in the context of divisible resources, most promi-nently in the cake-cutting literature; see (Brandt et al., 2016, Chapter 13) for an excellent survey. Thereis also a vast literature on connected (or contiguous) cake-cutting, spanning various notions of fairnessand economic efficiency (Stromquist, 1980; Su, 1999; Deng et al., 2012; Bei et al., 2012; Aumann et al.,2013; Aumann and Dombb, 2015; Segal-Halevi and Sziklai, 2018; Brânzei and Nisan, 2019; Goldberg et al.,2020). In particular, for equitability, it is known that for any given ordering of the agents, there exists aconnected equitable division of a cake consistent with the ordering (Cechlárová et al., 2013). Althoughno finite procedure can compute an exactly equitable division even without the connectedness con-straint (Procaccia and Wang, 2017), it is known that an ε -equitable connected division can be computedusing finite protocols (Cechlárová and Pillárová, 2012). Equitability has also been studied in combina-tion with other fairness notions. For example, while there always exists a connected equitable divisionthat is also proportional (Cechlárová et al., 2013), there might not exist a connected division that issimultaneously equitable and envy-free (Brams et al., 2006).For indivisible resources, the study of connected fair division has more recent ori-gins (Marenco and Tetzlaff, 2014; Bouveret et al., 2017; Suksompong, 2019). A number of fairnessnotions such as proportionality, envy-freeness, and maximin share have been examined in this modelwhen the resources are goods (Bouveret et al., 2017; Lonc and Truszczynski, 2020; Igarashi and Peters,2019; Bilò et al., 2019; Bei et al., 2021; Suksompong, 2019; Oh et al., 2019), chores (Bouveret et al., 2019),and mixed items involving both goods and chores (Aziz et al., 2019). A noteworthy result in thiscontext concerns the existence of allocations satisfying envy-freeness up to one good (EF1) when thenumber of agents is at most four (Bilò et al., 2019), or when agents have identical valuations (Bilò et al.,2019; Oh et al., 2019). As we observe in Remark 4, the latter result follows as a corollary of our mainalgorithmic result.For indivisible goods without the connectedness requirement, Gourvès et al. (2014) provided anefficient algorithm for achieving equitability up to any good . Subsequently, Freeman et al. (2019) studied(approximate) equitability along with Pareto optimality. Among other results, they showed that an EQ1and Pareto optimal allocation might fail to exist even with binary valuations, and provided efficientalgorithms for checking the existence of such allocations. By contrast, as we show in Theorem 2, theproblem becomes NP-complete when connectedness is also required. Let N = { a , a , . . . , a n } be a set of n ∈ N agents , and G = ( V, E ) be an undirected graph. Eachvertex v ∈ V of the graph G corresponds to an indivisible good (or item ) with m := | V | goods overall.A (connected) bundle is a set of goods S ⊆ V whose corresponding vertices induce a connected subgraphof G . We let C ( V ) ⊆ V denote the set of all connected subsets of V . Unless stated otherwise, we willassume that G is a path given by { v , v , . . . , v m } where { v i , v i +1 } ∈ E for i ∈ [ m − .A (connected) allocation A : N → C ( V ) assigns to each agent a i a connected bundle A ( a i ) ∈ C ( V ) such that no good is assigned to more than one agent. We will denote an allocation as an ordered tuple A = ( A , A , . . . , A n ) , where A i := A ( a i ) . An allocation is said to be complete if it does not leaveany good unassigned; that is, for any good v , there exists some agent a i such that v ∈ A i . A partial allocation is one that is not complete. Unless stated explicitly otherwise, the term ‘allocation’ will referto a complete allocation.The preferences of agent a i are specified by a valuation function u i : C ( V ) → N ∪ { } . We saythat the valuation functions are monotone if for any pair of connected bundles S, S ′ ∈ C ( V ) such that S ⊆ S ′ , we have u i ( S ) ≤ u i ( S ′ ) . The valuation functions are said to be additive if for each agent a i andeach bundle S ∈ C ( V ) , u i ( S ) := P v ∈ S u i ( { v } ) , where u i ( ∅ ) := 0 . Note that since all valuations arenon-negative, any additive valuation function is also monotone. We will assume throughout that the3aluations are additive (however, note that our algorithmic results apply to monotone, possibly non-additive valuations). For simplicity, we will write u i,j := u i ( { v j } ) . An n -tuple of valuation functions U = { u , . . . , u n } is called a valuation profile . We say that agents have binary (additive) valuations if u i,j ∈ { , } for all a i ∈ N and v j ∈ V . Fairness notions:
An allocation A is said to be• equitable ( EQ ) if for every pair of agents a i , a k ∈ N , the utilities of a i and a k for their respectivebundles are equal, that is, u i ( A i ) = u k ( A k ) ,• equitable up to one good ( EQ1 ) if for every pair of agents a i , a k ∈ N such that A k = ∅ , thereexists some good v ∈ A k such that u i ( A i ) ≥ u k ( A k \ { v } ) ,• envy-free ( EF) if for every pair of agents a i , a k ∈ N , u i ( A i ) ≥ u i ( A k ) , and• envy-free up to one good ( EF1 ) if for every pair of agents a i , a k ∈ N , u i ( A i ) ≥ u i ( A k \ { v } ) forsome v ∈ A k .The notions of EQ, EQ1. EF, and EF1 were formulated in the context of resource allocation byDubins and Spanier (1961), Freeman et al. (2019), Foley (1967), and Budish (2011), respectively. Notice that equitability and envy-freeness (and their corresponding relaxations) coincide whenagents have identical valuations (i.e., if u i = u k for every a i , a k ∈ N ) but are incomparable in gen-eral. Although our focus in this paper is on (approximate) equitability, some of our results also haveimplications for (approximate) envy-freeness. Efficiency notions:
An allocation A is said to be• Pareto optimal ( PO ) if for no other connected allocation B , we have u i ( B i ) ≥ u i ( A i ) for everyagent a i , with at least one of the inequalities being strict, and• non-wasteful ( NW ) if for any good v , there exists some agent a i such that v ∈ A i and u i ( { v } ) > . The utilitarian welfare of A is the sum of utilities of all agents in A , i.e., P a i ∈N u i ( A i ) , and the egali-tarian welfare of A is the utility of the least happy agent, i.e., min a i ∈N u i ( A i ) .Non-wastefulness and Pareto optimality are, in general, incomparable notions even when G is apath. However, for binary valuations, NW ⇒ PO ⇒ complete (since, for binary valuations, a non-wasteful allocation maximizes the utilitarian social welfare and is therefore Pareto optimal), and thereare simple examples where these implications are strict. Connected fair division problem:
The input to this problem is a tuple I = h G, N , U i consistingof a graph G , a set of agents N , and a valuation profile U . The goal is to determine whether I admits a connected allocation satisfying the desired notions of fairness and efficiency. Notice that if G is a clique,we recover the standard fair division model without the connectedness constraint. In this work, wewill exclusively focus on the case where G is a path graph. ( a, b ) -sparse instances: Given any ≤ a ≤ m and ≤ b ≤ n , we say that an instance with binaryvaluations is ( a, b ) -sparse if each agent approves at most a goods and each good is approved by at most b agents. Lipton et al. (2004) studied a weaker approximation of envy-freeness than EF1, but their algorithm is known to computean EF1 allocation. To make this notion well-defined, we will assume throughout that in any given instance, for every good there is at leastone agent with a non-zero value for it. This assumption is without loss of generality as our negative results (pertainingto computational hardness and non-existence) hold even under this assumption, and our positive results (algorithms andexistence results) do not need this assumption. Consider three goods v , v , v on a path and two agents with valuations u = (1 , , and u = (10 , , . Theallocation A := ( { v } , { v , v } ) is non-wasteful but is Pareto dominated by the (wasteful) allocation B := ( { v , v } , { v } ) . V U ′ V ′ . . . U p V p U ′ p V ′ p S C L S C R C L S C R . . . C Lp S p C Rp D D ′ . . . D p D ′ p Figure 1:
The instance used in the proof of Theorem 1.
Note that in the absence of the connectedness constraint, a non-wasteful allocation can be easily com-puted by assigning each good to an agent that has a positive value for it. By contrast, connectednessposes a substantial computational challenge even when we are only looking to satisfy non-wastefulness(without any fairness constraints), as the problem turns out to be NP-complete (Theorem 1).
Theorem 1.
Determining whether there exists a connected non-wasteful allocation is
NP-complete for apath and a (4 , -sparse binary valuations instance. To prove Theorem 1, we will show a reduction from a structured version of
Satisfiability called
Linear Near-Exact Satisfiability ( LNES ) which is known to be NP-complete (Dayal and Misra, 2019).An instance of
LNES consists of p clauses (where p ∈ N ) denoted as follows: C = { U , V , U ′ , V ′ , · · · , U p , V p , U ′ p , V ′ p } ∪ { C , · · · , C p } . We will refer to the first p clauses as the core clauses, and the remaining clauses as the auxiliary clauses.The set of variables consists of p main variables x , . . . , x p and p shadow variables y , . . . , y p .Each core clause consists of two literals and has the following structure: ∀ i ∈ [ p ] , U i ∩ V i = { x i } and U ′ i ∩ V ′ i = { ¯ x i } . Each main variable x i occurs exactly twice as a positive literal and exactly twice as a negative literal.The main variables only occur in the core clauses. Each shadow variable makes two appearances: asa positive literal in an auxiliary clause and as a negative literal in a core clause. Each auxiliary clauseconsists of four literals, each corresponding to a positive occurrence of a shadow variable.The LNES problem asks whether, given a set of clauses with the aforementioned structure, thereexists an assignment τ of truth values to the variables such that exactly one literal in every core clauseand exactly two literals in every auxiliary clause evaluate to true under τ . Proof. (of Theorem 1) Let φ be an instance of LNES . We will begin with a description of the reducedinstance.
Goods : Introduce one good for every core clause, denoted by U i , V i , U ′ i , and V ′ i , and two goodsfor every auxiliary clause, denoted by C Li and C Ri . We refer to these as core and auxiliary goods,respectively. We also introduce p dummy goods D , D ′ , . . . , D p , D ′ p as well as p + 1 separator goods S , S , . . . , S p . Thus, the total number of goods is m = 4 p + 2 p + 2 p + ( p + 1) = 9 p + 1 . The goodsare arranged as shown in Figure 1. Agents : For every main variable x i , we will introduce two agents a x i and a ¯ x i for the two literals;these are referred to as main agents of the positive and negative type, respectively. For every i ∈ [ p ] ,the agent a x i approves (i.e., values at ) the goods U i , V i , D i , D ′ i , while the agent a ¯ x i approves thegoods U ′ i , V ′ i , D i , D ′ i . We also introduce a shadow agent for every shadow variable. If y is a shadowvariable occurring in the core clause U i and auxiliary clause C j , then the shadow agent correspondingto y approves the goods U i , C Lj , and C Rj . The set of goods approved by y is analogously defined if itappears in the core clauses V i , U ′ i , or V ′ i . Finally, we introduce p + 1 separator agents s , s , . . . , s p such that for every i ∈ { , , . . . , p } , s i only approves the separator good S i . Thus, the total number5f agents is n = 2 p + 4 p + ( p + 1) = 7 p + 1 . Observe that the constructed instance is (4 , -sparse.We now turn to the proof of equivalence of the two instances. The Forward Direction.
Let τ be a satisfying assignment for the LNES instance. We will constructthe desired allocation as follows: For every i ∈ [ p ] , if the main variable x i evaluates to true (i.e., if τ ( x i ) = 1 ), then assign U i and V i to agent a x i , D i and D ′ i to agent a ¯ x i , and U ′ i and V ′ i to the (unique)shadow agents that approve these goods. Otherwise, if τ ( x i ) = 0 , then assign U ′ i and V ′ i to agent a ¯ x i , D i and D ′ i to agent a x i , and U i and V i to the (unique) shadow agents that approve these goods.Additionally, for every i ∈ { , . . . , p } , assign S i to the agent s i . Finally, for every i ∈ [ p ] , assign thegoods C Li and C Ri to the two shadow agents whose corresponding literals satisfy the auxiliary clause C i . The above allocation assigns each good to an agent that approves it and is therefore non-wasteful.It is also easy to see that the allocation is connected: The only agents that receive more than one goodunder this allocation are the main agents, and they always receive either two adjacent core goods ortwo adjacent dummy goods. The Reverse Direction.
We will now show how to recover an
LNES assignment given a connectedand non-wasteful allocation A for the fair division instance.Observe that due to non-wastefulness, each separator good is assigned to a unique separator agent,and the separator agents are not assigned any other goods. Thus, for every i ∈ { , , . . . , p } , A s i = { S i } . Similarly, the p dummy goods D , D ′ , . . . , D p , D ′ p must be allocated among at least p mainagents, which leaves at most p main agents for receiving the core goods. Furthermore, the separatorgoods prevent any shadow agent from getting more than one auxiliary good. Thus, the p auxiliarygoods are assigned to exactly p shadow agents, leaving the other p shadow agents for receiving thecore goods.Since each core good is approved by a unique shadow agent, at most p core goods can be allocatedamong shadow agents. Thus, the remaining p (or more) core goods should go to the main agents.However, due to non-wastefulness, a main agent cannot get more than two core goods. Overall, thismeans that one set of p main agents gets exactly two core goods each (the “lucky” agents), while theother set of p main agents gets two dummy goods each (the “unlucky” agents). Notice that the twomain agents corresponding to a main variable cannot both be lucky (since that would leave one ormore dummy goods unassigned), nor can both be unlucky (as that would create a similar violation forthe core goods).This brings us to a natural way of deriving an assignment τ from the allocation A . If the main agentof the positive (respectively, negative) type is unlucky, then we let τ ( x i ) = 0 (respectively, τ ( x i ) = 1 ).Furthermore, if A allocates a core good to a shadow agent, then the corresponding shadow variable isset to , while shadow variables corresponding to shadow agents who receive auxiliary goods are setto . Note that exactly p of the p shadow variables are set to . It can be verified that τ is indeed asatisfying assignment.Notice that the allocation obtained in the forward direction in the proof of Theorem 1 is EQ1 and EF1,and the argument for the reverse direction is driven only by non-wastefulness. Thus, we also obtainhardness results for EQ1+NW and EF1+NW allocations. Additionally, for binary additive valuations, anallocation is non-wasteful if and only if its utilitarian welfare is at least m . These observations establishthe hardness of a number of related problems. Corollary 1.
Checking the existence of a connected allocation that is (a)
EQ1 and NW , (b) EF1 and NW , (c) EQ1 and has utilitarian welfare at least m , or (d) EF1 and has utilitarian welfare at least m is NP-complete for a path and a (4 , -sparse binary valuations instance. emark . The hardness result in Corollary 1 can be extended to multiplicative approximations of EQ1and EF1. Given any α ∈ [0 , , an allocation A is said to satisfy α -EQ1 (respectively, α -EF1) if forevery pair of agents a i , a k ∈ N such that A k = ∅ , there exists some good v ∈ A k such that u i ( A i ) ≥ α · u k ( A k \ { v } ) (respectively, u i ( A i ) ≥ α · u i ( A k \ { v } ) ). The reasoning is similar: The allocationin the forward direction is EQ1 as well as EF1, and hence also α -EQ1 and α -EF1. The argument in thereverse direction only uses non-wastefulness, and therefore vacuously holds for α -EQ1 (or α -EF1). Asa result, we obtain that for any rational α ∈ [0 , , it is NP-complete to determine the existence of aconnected α -EQ1 (or α -EF1) allocation that is non-wasteful or has utilitarian welfare at least m .A straightforward adaptation of the construction in Theorem 1 also gives us the following: Theorem 2.
Checking the existence of a connected allocation that is (a)
EQ1 and PO , (b) EF1 and PO , (c) EQ1 and has egalitarian welfare at least , or (d) EF1 and has egalitarian welfare at least is NP-complete for a path and a (6 , -sparse binary valuations instance. The proof of Theorem 2 is presented in the Appendix A.Recently, Igarashi and Peters (2019, Theorem 7) have shown NP-hardness of checking the existenceof a connected EF1+PO allocation of a path even for binary valuations. Their construction involvesitems that are valued by all agents, thus requiring O ( n ) sparsity. By contrast, our result in Theorem 2shows hardness even for O (1) sparse instances. Finally, we note that the proof of Theorem 1 can alsobe adapted to show NP-hardness for egalitarian or utilitarian-optimal EQ1 allocations of chores (therelevant transformation is u ′ i,j = u i,j − ). The intractability results in the previous section prompt us to relax the efficiency requirement in searchof positive results, and ask the following question: Does there always exist a connected and complete
EQ1 allocation of a path?A natural approach towards this question is to start with a connected and exactly equitable divisionin a cake-cutting instance derived by relaxing the indivisibility constraint (such divisions are guaran-teed to exist (Cechlárová et al., 2013; Aumann and Dombb, 2015; Chèze, 2017)). The fractional cakedivision could then be rounded to obtain a connected and approximately equitable allocation of indivis-ible goods. Unfortunately, there exist instances where every rounding of the fractional cake divisionfails to satisfy EQ1. An alternative approach is to work directly with the indivisible goods instance. For a path graph,any connected allocation can be naturally associated with a left-to-right ordering of agents, say σ .We call a connected (partial) allocation σ -consistent if it assigns connected bundles from left to rightaccording to σ . Suksompong (2019) has shown that there is a polynomial-time local search algo-rithm that, for any fixed ordering σ of agents, finds a connected, complete, σ -consistent, and ap-proximately equitable allocation. Specifically, his algorithm computes a u max -EQ allocation, where u max := max a i ∈N ,v ∈ V u i ( { v } ) is the highest valuation any agent has for any good, and an allocation A is u max -EQ if for every a i , a k ∈ N , we have | u i ( A i ) − u k ( A k ) | ≤ u max . Similar approximations have been studied in the context of envy-freeness up to any good (EFX) (Plaut and Roughgarden,2020; Amanatidis et al., 2020). For negatively-valued items (or chores), an allocation is said to satisfy EQ1 if for every pair of agents a i , a k ∈ N suchthat A i = ∅ , there exists a chore v ∈ A i such that v i ( A i \ { v } ) ≥ v k ( A k ) (Freeman et al., 2020). Consider an instance with seven goods v , . . . , v and three agents with identical valuations u = (1 , , , , , , . Any connected and equitable division assigns v , . . . , v to one agent and equally divides v between the other two. In anyrounding, some agent will get an empty bundle, thus violating EQ1. u max -EQ is a strictly weaker guarantee than EQ1, and there exist instances where Suk-sompong’s algorithm fails to compute an EQ1 allocation (even though such an allocation exists). Thus,this approach, too, does not resolve the existence of EQ1 and complete allocations. Moreover, this al-gorithm could fail to satisfy standard criteria of economic efficiency . Given this limitation, Suksompong(2019) posed the computation of ‘approximate equitable allocations with non-trivial welfare guarantees’as an open problem.We address this gap by providing a polynomial-time algorithm for computing a connected, com-plete, and EQ1 allocation (Theorem 3). Our algorithm also provides the following economic efficiencyguarantee: For any given agent ordering σ , our algorithm returns a connected, σ -consistent, and EQ1allocation whose egalitarian welfare is the highest among all connected and σ -consistent allocations.In other words, a connected and egalitarian-optimal allocation for any fixed ordering of the agents is,without loss of generality, fair (i.e., EQ1) and efficiently computable. Theorem 3.
There is a polynomial-time algorithm for computing a connected, complete, and
EQ1 alloca-tion of a path consistent with a given ordering of agents. Furthermore, this allocation is egalitarian-optimalamong all connected allocations consistent with the given ordering.
Note that the strong existence guarantee of Theorem 3 cannot be extended to EQ1 and Pareto opti-mal allocations. Indeed, consider an instance with five goods and three agents where u = (1 , , , , , u = (0 , , , , , and u = (0 , , , , . For σ = (1 , , , the unique connected, σ -consistent, andPareto optimal allocation is ( { v } , { v } , { v , v , v } ) which violates EQ1. Description of the algorithm:
Let σ := ( a , a , . . . , a n ) . Our algorithm (see Algorithm 1) consistsof three phases.In Phase 1, the algorithm computes the optimal egalitarian welfare θ for σ -consistent allocations.To compute this value, the algorithm starts with a preprocessed list L = ( u , u , . . . ) containing alldistinct realizable utility values of any agent for any connected bundle, where u := 0 < u < u and soon. (The list L is of length O ( nm ) since the number of distinct connected bundles in a path is O ( m ) .)In round k , the algorithm checks whether there exists a connected and σ -consistent partial allocationwith egalitarian welfare u k +1 . To do this, the algorithm starts from the leftmost available good anditeratively assigns minimal connected bundles to the agents a , a , . . . such that each agent gets autility of at least u k +1 ; here, minimal refers to cardinality-wise smallest bundle. If a feasible partialallocation exists, the algorithm updates its ‘guess’ of the achievable egalitarian welfare to θ = u k +1 and moves to round k + 1 . Otherwise, it fixes θ = u k and moves to Phase 2. Thus, for the instance inFigure 2, the partial allocation in round ( θ = 0 ) is ( { v } , { v , v } , { v } ) , and that in round ( θ = 1 ) is ( { v , v } , { v , v } , { v , v , v , v } ) . In round , the algorithm encounters infeasibility, so it fixes θ = 2 .In Phase 2, the algorithm searches for a θ - unsafe agent . Given any θ = u k , we say that agent a i is θ - safe if there exists a connected and σ -consistent (partial) allocation in which each of a , a , . . . , a i gets a utility of at least u k +1 , and each of a i +1 , . . . , a n gets a utility of at least u k . A θ -unsafe agent isone that is not θ -safe (see Figure 2). Note that a θ -unsafe agent must exist since we know from Phase1 that an egalitarian welfare of u k +1 is not possible. The procedure in Phase 1 can be easily adapted tocompute the leftmost θ -unsafe agent, say a i . Having found a i , the algorithm now fixes the assignmentsof its predecessors a , . . . , a i − (but not a i ) by starting from the leftmost available good and iterativelyassigning each agent a minimal connected bundle worth at least u k +1 . The algorithm now moves toPhase 3.In Phase 3, the algorithm finalizes the assignments of the remaining agents via a right-to-left scanof the path G . Specifically, starting from the rightmost available good, the algorithm moves leftwards Consider the instance in Footnote 6 where u max = 12 . Starting with the allocation A := ( {∅} , { v , . . . , v } , { v } ) ,Suksompong’s local search algorithm immediately returns A as the output since it is u max -EQ, even though it violates EQ1.Observe that the allocation B := ( { v , v , v } , { v , v , v } , { v } ) is EQ1 and has a higher egalitarian welfare. LGORITHM 1:
Algorithm for finding a connected and complete EQ1 allocation.
Input:
An instance I = h G, N , Ui and an ordering of agents σ . Output:
A connected allocation A . ⊲ Phase : Compute the optimal egalitarian welfare θ k ← θ ← u k ⊲ Initialize the guess for optimal egalitarian welfare while k ≤ n ( m + 1) do Starting from the leftmost available good, move rightward along G and tentatively assign a minimalconnected bundle worth at least u k +1 to each successive agent in σ . if the assignment in Line 4 is infeasible then Exit while-loop and start Phase 2. else θ ← u k +1 ⊲ Update the guess k ← k + 1 ⊲ Update k⊲ Phase : Find a θ -unsafe agent via a left-to-right scan A ← ( ∅ , . . . , ∅ ) i ← while i ≤ n do if there exists a σ -consistent partial allocation that is identical to A for the agents a , . . . , a i − , and, inaddition, assigns connected bundles worth at least u k +1 to a i and worth at least θ = u k to each of a i +1 , . . . , a n then A i ← the minimal connected bundle worth at least u k +1 to agent a i starting from the leftmostavailable good. G ← G \ A i ⊲ Update the set of remaining goods i ← i + 1 else Exit while-loop and start Phase 3. ⊲ a i is the leftmost θ -unsafe agent ⊲ Phase : Finalize the remaining assignments via a right-to-left scan k ← n while k > i do A k ← the minimal connected bundle worth at least θ = u k to agent a k starting from the rightmostavailable good. G ← G \ A k ⊲ Update the set of remaining goods k ← k − A i ← G ⊲ Assign all remaining goods to the θ -unsafe agent a i return A v a : a : a : v v v v v v v u = 3 u = 2 u = 2 a is -safe θ = u k = 2 u k +1 = 3 u = 3 u = 3 u < a is - unsafe Figure 2:
Illustrating the notion of θ -unsafe agent on an instance with binary valuations. For θ = u k = 2 and u k +1 = 3 , agent a is θ -safe because there exists a partial allocation in which a ’s utility is at least u k +1 and thatof each of its successors is at least u k . Agent a is θ -unsafe because giving a utility of at least u k +1 to both a and a necessarily involves a ’s utility being less than u k . G and iteratively assigns minimal connected bundles worth at least u k to the agents in the reverseorder a n , a n − , and so on. Upon encountering a i for the second time, the algorithm assigns to it all theremaining goods, and returns the final allocation as the output. The proof of Theorem 3 follows. Proof. (of Theorem 3) We will show that the above algorithm (Algorithm 1) satisfies the desired prop-erties.Let us start with the running time analysis of the algorithm assuming that agents have additivevaluations (later in Remark 2, we will provide a similar analysis for general monotone valuations). Sincethere are O ( m ) possible connected bundles for each of the n agents, the computation of L requirescomputing the utility of O ( nm ) bundles by adding up the individual utilities of the constituent goods.This amounts to a total running time of O ( nm log( u max )) for the preprocessing phase. Further, weassume that the agents’ utilities for all possible bundles are cached so as to facilitate constant timeaccess in the remainder of the algorithm.The total running time for Phase 1 is O ( nm ) , since there are at most n ( m + 1) iterations of thewhile-loop and each iteration involves scanning at most m goods. Phase 2 involves at most n iterationsof the while-loop, and each iteration requires assigning connected bundles to all agents from left toright, which takes O ( m ) time. Thus, the total running time for Phase 2 is O ( nm ) . In Phase 3, eachgood is considered at most once during the right-to-left scan, resulting in a running time of O ( m ) .Thus, overall, the algorithm takes O ( nm + nm log( u max )) time.The allocation A returned by the algorithm is complete because all leftover goods are allocated inthe last step, and is σ -consistent because this property is maintained by the algorithm at every step.Furthermore, A is also connected since the algorithm assigns connected bundles to a , . . . , a i − fromleft to right and to a n , . . . , a i +1 from right to left. (The feasibility of the right-to-left assignment isguaranteed by the fact that a i − is θ -safe, as a i is leftmost θ -unsafe agent.) Since G is a path, the set ofleftover goods assigned to a i in Phase 3 is also connected.We will now argue that A is egalitarian-optimal among all connected and σ -consistent allocations.First, observe that the value θ = u k fixed at the end of Phase 1 is indeed the optimal egalitarian welfareof any connected and σ -consistent allocation. (Otherwise, by monotonicity of valuations, there mustexist a connected and σ -consistent allocation with egalitarian welfare u k +1 or higher in which agentsreceive minimal bundles. This, however, would contradict the infeasibility encountered for θ = u k +1 .)Next, we will show that the egalitarian welfare of A is equal to u k , which will establish egalitarian-optimality. Indeed, each of a , . . . , a i − gets a utility of at least u k +1 > u k in Phase 2, and each of a n , . . . , a i +1 gets a utility of at least u k in Phase 3. The utility of a i for its assigned bundle is exactly u k because of the following two reasons: First, a i ’s utility is at least θ = u k since a i − is θ -safe (recall that a i is the leftmost θ -unsafe agent). Second, since a i is θ -unsafe, assigning a bundle worth at least u k +1 to a i (and each of its predecessors a , . . . , a i − ) would imply that one of its successors a i +1 , . . . , a n gets utility strictly below u k , which contradicts the assignments in Phase 3. Thus, a i ’s utility must bestrictly below u k +1 , and hence, equal to u k = θ .Finally, to prove that A is EQ1, notice that if the utility of an agent is strictly greater than u k (inparticular, each of a , . . . , a i − gets a utility at least u k +1 > u k ), then by minimality of bundles, theremust exist a boundary good whose removal results in the agent’s residual utility being strictly below u k +1 , and therefore less than or equal to u k . Since each agent gets a utility at least u k , A must beEQ1.We observe that the running time of Algorithm 1 can be improved to O ( nm ) via following modi-fications: In the preprocessing step, as before, we go through all possible O ( nm ) bundles. We cachethese bundles, and compute u max which amounts to the running time of O ( nm ) for this step. Next,in Phase 1, we use binary instead of the linear search to find the optimal egalitarian welfare θ . Withthese modifications, Phase 1 runs in time O ( m log mn ) . In Phase 2, the θ -unsafe agent can be found in10 ( m ) time with a combination of left-to-right scan that tentatively assigns bundles worth u k +1 and aright-to-left scan that assigns bundles worth u k . Finally, Phase 3 runs takes O ( m ) time as before. Remark . Note that the algorithm in Theorem 3 and the analysis of its correctness only use the mono-tonicity of valuations, and therefore the result extends to non-additive utilities. The running time anal-ysis in this case relies on the existence of a valuation oracle that, given a connected bundle, returns theagent’s utility for that bundle. Since the number of distinct connected bundles in a path is O ( m ) , after O ( nm ) valuation queries, each agent’s value for every connected bundle is available to the algorithm.The rest of the analysis is identical to that in Theorem 3. Remark . Another relevant implication is that our algorithm can be easily adapted for negative val-uations to obtain the efficient computation of connected EQ1 allocations for chores . The latter resultprovides a tractable alternative to a recent result showing NP-hardness for connected and exactly eq-uitable chore allocations (Bouveret et al., 2019).
Remark . Bilò et al. (2019) and Oh et al. (2019) have independently shown that when agents have iden-tical monotone valuations, a connected EF1 allocation of a path can be efficiently computed. Since EF1and EQ1 coincide for identical valuations, our result in Theorem 3 implies this result as a corollary.Additionally, we note that although the algorithm of Bilò et al. (2019) and its analysis are presentedfor identical valuations, a natural extension of their algorithm for general valuations can be used toderive an alternative proof of Theorem 3.
The existence result in Theorem 3 is quite general, since it applies to any fixed ordering of agents andany monotone valuations instance, and reconciles fairness (i.e., EQ1) with a weak form of economicefficiency (i.e., completeness). On closer inspection, though, we find that it implies an even strongerexistence result. Specifically, given an agent ordering σ , let A σ denote the set of all connected, σ -consistent, EQ1 and complete allocations for the given instance. From Theorem 3, we know that A σ isnon-empty. Furthermore, since there are only finitely many allocations, there must exist an allocationin A σ that is not Pareto dominated by any other allocation in A σ . We call this property PO*.Formally, given an agent ordering σ , we say that allocation A is PO* if it is connected, σ -consistent,complete, and EQ1, and no other connected, σ -consistent, complete, and EQ1 allocation Pareto domi-nates A . From the aforementioned argument, it follows that a PO* allocation always exists.Intriguingly, while Algorithm 1 can be used to establish the existence of a PO* allocation even forgeneral monotone valuations, it can fail to return such an allocation even for binary additive valua-tions. Indeed, consider an instance with five goods v , . . . v and three agents with valuations u =(1 , , , , , u = (0 , , , , , and u = (0 , , , , . Given the ordering σ = (1 , , , Algorithm 1computes a σ -consistent and EQ1 allocation ( { v , v } , { v , v } , { v } ) with utility profile (1 , , , whichis Pareto dominated by another σ -consistent and EQ1 allocation ( { v } , { v , v } , { v , v } ) with utilityprofile (1 , , .Thus, for a given agent ordering σ , PO* is stronger than EQ1+completeness as in this case, theformer implies the latter. We note that PO* with respect to an ordering σ could be weaker (i.e., Paretodominated) than an EQ1+complete allocation with respect to a different ordering σ ′ .This motivates the following natural question: Given an ordering σ , can a PO* allocation be ef-ficiently computed? While we are unable to settle this question for general monotone valuations, inTheorem 4 we show that a variant of Algorithm 1 efficiently computes a PO* allocation for binaryadditive valuations. Theorem 4.
Given an instance with binary additive valuations and any agent ordering σ , a connected, σ -consistent, EQ1 , and PO * allocation of a path can be computed in polynomial time. The proof of Theorem 4 is presented in the Appendix B.11
Structured Preferences
In this section, we will explore a different avenue for circumventing the intractability associated withnon-wasteful EQ1 allocations. Unlike in Theorem 3 where we relaxed the efficiency requirement, thistime we will instead assume that agents have structured preferences. In particular, we will focus on binary extremal valuations wherein for each agent a i , either there exists ℓ i ∈ [ m ] such that u i,j = 1 forall j ∈ { , . . . , ℓ i } and otherwise (i.e., a i is left-extremal ), or there exists r i ∈ [ m ] such that u i,j = 1 for all j ∈ { r i , . . . , m } and otherwise (i.e., a i is right-extremal ). Similar domain restrictions havebeen previously considered in the context of voting problems (Elkind and Lackner, 2015). Theorem 5.
There is a polynomial-time algorithm that, given an instance with binary extremal andadditive valuations, returns a connected, non-wasteful, and
EQ1 allocation whenever such an allocationexists.Proof.
We will show that the desired allocation, if it exists, can be obtained by concatenating the so-lutions from two subproblems, one on a purely left-extremal and the other on a purely right-extremalsubinstance.Suppose there exists a connected, non-wasteful (NW), and EQ1 allocation A . Let σ denote the agentordering under A . By relabeling the agents, we have that σ = ( a , . . . , a n ) . We claim that without lossof generality, all left-extremal agents precede all right-extremal agents in σ . Indeed, if there is a pairof adjacent agents a i , a i +1 where a i is right-extremal and a i +1 is left-extremal, then by an exchangeargument we can obtain another connected, non-wasteful, and EQ1 allocation B where such a violationdoes not occur. Specifically, by swapping the bundles of a i and a i +1 , we maintain connectedness andnon-wasteful. Additionally, for binary additive valuations, non-wastefulness implies that the utility ofan agent is equal to the cardinality of its bundle. Therefore, swapping bundles results in swapping theutility values of a i and a i +1 , which means that the old and new allocations have identical utility profiles(up to relabeling). Thus, allocation B must also satisfy EQ1.Let v j ∈ V be such that the set V L := { v , . . . , v j } is allocated among the left-extremal agentsand V R := { v j +1 , . . . , v m } is allocated among the right-extremal agents in A . Then, the subinstancerestricted to V L only has left-extremal valuations and admits a connected, non-wasteful, and EQ1 allo-cation (indeed, the restriction of A to V L satisfies these properties). A similar implication holds for thepurely right-extremal subinstance V R . Therefore, it suffices to provide a polynomial-time algorithmfor checking the existence of a connected, non-wasteful, and EQ1 allocation in a binary left-extremalinstance. Notice that the same algorithm can be used for the right-extremal subinstance via an easy‘mirror transformation’. If both subinstances admit desired allocations, then the concatenated alloca-tion is clearly connected and non-wasteful in the original instance. By checking this allocation for EQ1,we obtain the desired algorithm for the original instance. Thus, in rest of the proof, we will focus onlyon left-extremal valuations.Let A ′ denote the restriction of allocation A to the left-extremal subinstance, and let n ′ and m ′ correspondingly denote the number of agents and items, respectively. Since A ′ is non-wasteful andEQ1 and the valuations are binary, the minimum and maximum utilities under A ′ must be ⌊ m ′ n ′ ⌋ and ⌈ m ′ n ′ ⌉ , respectively. That is, an agent is either a ‘floor’ or a ‘ceiling’ agent. By an exchange argument, itcan be shown that for any pair of left-extremal agents a i , a k such that i < k , we have ℓ i ≤ ℓ k (i.e., a i ’sinterval finishes before a k ’s) without loss of generality. Similarly, it holds that the floor agents precedethe ceiling agents (here, the exchange argument transfers a boundary item).Let n ′ f and n ′ c denote the number of floor and ceiling agents, respectively. Thus, n ′ f and n ′ c are theunique pair of non-negative integers satisfying the equations n ′ f + n ′ c = n ′ and n ′ f · ⌊ m ′ n ′ ⌋ + n ′ c · ⌈ m ′ n ′ ⌉ = m ′ . (If m ′ = kn ′ for some k ∈ N , then n ′ f = n ′ and n ′ c = 0 .) The desired algorithm considers the agentsin the order in which their intervals finish, and constructs an allocation as follows: Starting from theleftmost available good, the algorithm assigns a connected bundle of ⌊ m ′ n ′ ⌋ goods to each of the first n ′ f ⌈ m ′ n ′ ⌉ goods to each of the next n ′ c agents. If this allocation is non-wasteful and EQ1, then the algorithm reports YES and returns the said allocation, otherwise it reportsNO. We initiated the study of EQ1 allocations under connectedness constraints. The pursuit of connectedEQ1 allocations satisfying non-trivial efficiency guarantees resulted in computational hardness. Thisresult motivated the exploration of two avenues for tractability: relaxing the efficiency requirementand assuming structured preferences. Some of our results found broader applicability to other fairnessnotions (e.g., EF1) and negatively-valued items.Going forward, it would be very interesting to explore the domain of binary intervals without theextremal structure in search of tractability results. Another relevant direction could be to map theintractability frontier for binary valuations in terms of ( a, b ) -sparsity . Our results establish hardnessof a number of problems even under (4 , -sparsity. On the other hand, (1 , b ) -sparse instances areefficiently solvable for any b . Resolving the complexity of intermediate cases is a natural next step.Finally, extensions to general graphs (Bouveret et al., 2017) or settings with mixed items involving goodsas well as chores (Aziz et al., 2019) could also be of interest. Acknowledgments
We thank the anonymous reviewers for helpful comments and suggestions. NM is supported by theSERB ECR grant ECR/2018/002967, Computational Aspects of Social Choice: Theory and Practice. RVacknowledges support from ONR
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Let us recall the statement of Theorem 2.
Theorem 2.
Checking the existence of a connected allocation that is (a)
EQ1 and PO , (b) EF1 and PO , (c) EQ1 and has egalitarian welfare at least , or (d) EF1 and has egalitarian welfare at least is NP-complete for a path and a (6 , -sparse binary valuations instance. We will start by discussing the proof of part (a) of Theorem 2, followed by that of part (c) whichuses the same construction. Parts (b) and (d) use a slightly different construction, and their proofs willbe presented subsequently.
Proof. (of part (a)) We will show a reduction from
Linear Near-Exact Satisfiability ( LNES ) and ourconstruction will be similar to that of Theorem 1. Recall that an instance of
LNES consists of p clauses(where p ∈ N ) denoted as follows: C = { U , V , U ′ , V ′ , · · · , U p , V p , U ′ p , V ′ p } ∪ { C , · · · , C p } . We will refer to the first p clauses as the core clauses, and the remaining clauses as the auxiliary clauses.The set of variables consists of p main variables x , . . . , x p and p shadow variables (our notation forthe shadow variables will differ slightly from that used in Theorem 1).Each core clause consists of two literals and has the following structure: ∀ i ∈ [ p ] , U i ∩ V i = { x i } and U ′ i ∩ V ′ i = { ¯ x i } . Each main variable x i occurs exactly twice as a positive literal and exactly twice as a negative literal.The main variables only occur in the core clauses. Each shadow variable makes two appearances: as apositive literal in an auxiliary clause and as a negative literal in a core clause. For i ∈ [ p ] , we will let p i , r i , q i , and s i denote the shadow variables that appear (as negative literals) in the core clauses U i , V i , U ′ i and V ′ i , respectively. That is, U i := ( ¯ p i ∧ x i ) , V i := ( ¯ r i ∧ x i ) , U ′ i := ( ¯ q i ∧ ¯ x i ) , and V ′ i := ( ¯ s i ∧ ¯ x i ) .Each auxiliary clause consists of four literals, each corresponding to a positive occurrence of a shadowvariable.The LNES problem asks whether, given a set of clauses with the aforementioned structure, thereexists an assignment τ of truth values to the variables such that exactly one literal in every core clauseand exactly two literals in every auxiliary clause evaluate to true under τ . Construction of the reduced instance.
Let φ be an instance of LNES. We will begin with thedescription of the reduced instance. Goods:
For every i ∈ [ p ] , we introduce one good for every core clause denoted by U i , V i , U ′ i , V ′ i ,and six goods for every auxiliary clause denoted by C L i , C L i , S i , S i , C R i , C R i . We refer to U i , V i , U ′ i , V ′ i as the core goods, C L i , C L i , C R i , C R i as the auxiliary goods, and S i , S i as the separator goods .Next, we introduce two goods for each shadow variable, i.e., corresponding to each of p i , q i , r i , s i , weintroduce the following shadow goods: p i , p i , r i , r i , q i , q i , s i , s i . Finally, we introduce p dummy goods denoted by D , D ′ , . . . , D p , D ′ p , two additional separator goods S , S , and three special goods S , S , S . Thus, the total number of goods is m = 4 p + 6 p + 8 p + 2 p + 2 + 3 = 20 p + 5 . The goodsare arranged as shown in Figure 3. Agents:
For every main variable x i , we will introduce two agents a x i and a ¯ x i for the two literals;these are referred to as main agents of the positive and negative type, respectively. For every i ∈ [ p ] , theagent a x i approves (i.e., values at ) the goods U i , V i , D i , D ′ i , while the agent a ¯ x i approves the goods U ′ i , V ′ i , D i , D ′ i . We also introduce a shadow agent for every shadow variable. If p i is a shadow variableoccurring in core clause U i and auxiliary clause C j , then the corresponding shadow agent p i approves17 , p , p , r , r , V , U ′ , q , q , s , s , V ′ , · · · , U p , p p , p p , r p , r p , V p , U ′ p , q p , q p , s p , s p , V ′ p (Core and shadow goods) S , S , C L , C L , S , S , C R , C R , · · · , C L p , C L p , S p , S p , C R p , C R p (Separator and auxiliary goods) D , D ′ , D , D ′ , · · · , D p , D ′ p , S , S , S (Dummy and special goods) Figure 3:
The instance used in the proof of part (a) of Theorem 2. The path graph is such that the goods in the toprow are to the left of those in the middle row, which are to the left of those in the bottom row. the shadow goods p i , p i and the auxiliary goods C L j , C L j , C R j , C R j . The valuations of the othershadow agents r i , q i , s i are defined analogously. Next, we introduce p + 1 separator agents t , . . . , t p such that for every i ∈ { } ∪ [ p ] , t i approves two separator goods S i , S i . Lastly, we introduce specialagent a s that approves the special goods S , S , S .This completes the construction of our reduction. Notice that the constructed instance is (6 , -sparse . Before presenting the proof of equivalence, we will establish in Lemma 1 that each agent (exceptfor the special agent) has a utility of under any EQ1 and Pareto optimal allocation. Lemma 1.
In any
EQ1 + PO allocation, the utility of the special agent a s is equal to and that of everyother agent is equal to .Proof. (of Lemma 1) Notice that in any Pareto optimal allocation A , the special goods S , S , S must beallocated to the special agent a s . This is because these goods lie at the end of the path and are uniquelyvalued by a s , and therefore any allocation A ′ that does not assign these goods to a s can be shown to bePareto dominated by another allocation that is identical to A ′ except for the assignment of the specialgoods to the special agent. Therefore, the utility of a s under Pareto optimal allocation must be equalto (recall that a s does not value any good other than the special goods).Now let A denote any EQ1 and Pareto optimal allocation. Since the utility of the special agent in A is equal to , EQ1 requires that the utility of every other agent in A is at least .Since each separator agent t , t , . . . , t p approves exactly two goods, it must be that for every i ∈{ , , . . . , p } , the separator goods S i , S i are assigned to t i in A . Furthermore, since the separator goods S i , S i are placed next to each other on the path and these are the only goods approved by t i , we canassume, without loss of generality, that these are the only goods assigned to t i .Now consider a shadow agent p i that appears in the core clause U i and the auxiliary clause C j . Thus, p i approves two shadow goods p i , p i and four auxiliary goods C L j , C L j , C R j , C R j . Note that p i cannotreceive more than two approved goods; if it does, then by connectedness constraint, its bundle shouldnecessarily include separator goods whose assignment has already been fixed. Thus, each shadow agent p i (analogously q i , r i , s i ) will have a utility of exactly in A .A similar argument shows that for any i ∈ [ p ] , the main agent of positive (or negative) type a x i (or a ¯ x i ) will have a utility of at most since all such agents approve two core goods and two dummy goods.We therefore have that in any EQ1 and Pareto optimal allocation, all agents other than the special agentachieve a utility of exactly . This completes the proof of Lemma 1. The Forward Direction.
Given a satisfying assignment τ for LNES , we will construct the desiredallocation as follows:• Allocate the special goods S , S , S to the special agent a s .• For each i ∈ { , , . . . , p } , the separator agent t i receives the separator goods S i and S i .18 If τ ( x i ) = 1 , then allocate { U i , p i , p i , r i , r i , V i } to agent a x i and { D i , D ′ i } to agent a ¯ x i . Inaddition, allocate { U ′ i , q i , q i } to q i , and { s i , s i , V ′ i } to s i . Recall that q i and s i are the shadowvariables that appear as negated literals in the core clauses U ′ i and V ′ i , respectively, along with ¯ x i .Otherwise, if τ ( x i ) = 0 , then allocate { U ′ i , q i , q i , s i , s i , V ′ i } to agent a ¯ x i and { D i , D ′ i } to agent a x i . In addition, allocate { U i , p i , p i } to p i , and { r i , r i , V i } to r i .• Finally, for every j ∈ [ p ] , allocate the sets { C L j , C L j } and { C R j , C R j } to the two shadow agentswhose corresponding literals satisfy the auxiliary clause C j .Observe that each good is assigned to exactly one agent in the aforementioned allocation. Fur-thermore, each agent’s bundle is connected; in particular, each shadow agent either receives a set ofadjacent core and shadow goods (if the corresponding shadow variable evaluates to false under τ ), ora set of adjacent auxiliary goods (if it evaluates to true).It is easy to verify that the utility of the special agent is equal to , and that of every other agent isequal to . Thus, the allocation is EQ1.We will now argue that the above allocation, say A , is Pareto optimal. Suppose, for contradiction,that another allocation A ′ Pareto dominates A . Since the special agent and each separator agent re-ceives all of its approved goods under A , the utilities of these agents under A and A ′ must be equal.Furthermore, if a main agent has a strictly higher utility under A ′ , then by the connectedness constraint,its bundle must contain a separator good, which leads to an infeasible assignment since these goodsare necessarily allocated to the separator agents. A similar argument shows that a shadow agent, too,cannot receive a higher utility under A ′ . Therefore, A must be Pareto optimal. The Reverse Direction.
We will now show how to recover an
LNES assignment given a connectedEQ1 and Pareto optimal allocation, say A .Since A is EQ1 and Pareto optimal, we know from Lemma 1 that the special agent receives threeapproved goods and every other agent receives two approved goods under A . Thus, in particular, forevery i ∈ { , , . . . , p } , the separator goods S i , S i are allocated to the separator agent t i . Along withthe connectedness constraint, this implies that for every i ∈ [ p ] , at least one of the main agents a x i or a ¯ x i will achieve a utility of by either receiving the interval U i , p i , p i , r i , r i , V i or U ′ i , q i , q i , s i , s i , V ′ i .This, in turn, forces at least one pair of shadow agents—either { p i , r i } or { q i , s i } —to obtain their utilitiesfrom the auxiliary goods.We will now show that exactly one of these two pairs of agents derive their utility from the shadowgoods, while the other pair meets the utility requirement though the auxiliary goods. Indeed, sincethere are p auxiliary goods (corresponding to p auxiliary clauses), at most p shadow agents canobtain the desired utility from the auxiliary goods. Therefore, for every i ∈ [ p ] , exactly one pair ofshadow agents—either { p i , r i } or { q i , s i } —are assigned shadow goods, while the other pair receivesauxiliary goods. Note that this observation also shows that for every i ∈ [ p ] , exactly one out of a x i or a ¯ x i is assigned the dummy goods { D i , D ′ i } .Overall, we have that one set of p main agents gets exactly two core goods each (we will refer themas the “lucky” agents), while the other set of p main agents gets two dummy goods each (the “unlucky”agents). Notice that the two main agents corresponding to a main variable cannot both be lucky, norcan both be unlucky due to the argument presented earlier.This brings us to a natural way of deriving an LNES assignment τ from the allocation A . If themain agent of the positive (respectively, negative) type is unlucky, then we let τ ( x i ) = 0 (respectively, τ ( x i ) = 1 ). Furthermore, if A allocates a core good to a shadow agent, then the corresponding shadowvariable is set to , while shadow variables corresponding to shadow agents who receive auxiliary goodsare set to . Note that exactly p of the p shadow variables are set to under this assignment and19 , p , p , r , r , V , U ′ , q , q , s , s , V ′ , · · · , U p , p p , p p , r p , r p , V p , U ′ p , q p , q p , s p , s p , V ′ p (Core and shadow goods) S , S , C L , C R , · · · , C Lp , C Rp , D , D , D , · · · , D p , D p , D p (Separator and auxiliary goods followed by the dummy goods) Figure 4:
The instance used in proof of part (b) of Theorem 2. The path graph is constructed such that the goods inthe top row are to the left of those in the bottom row. there are no conflicting assignments, implying that τ is indeed a valid solution to the LNES instance.This completes the proof of part (a) of Theorem 2.
Proof. (of part (c)) To prove part (c), we first observe that the argument in the forward direction remainsthe same as in part (a), since the allocation constructed in the proof is EQ1 and satisfies the desiredegalitarian welfare condition.In the reverse direction, it is possible that under the given allocation, say A , the special agent a s nolonger receives all three special goods. However, since the egalitarian welfare of A is at least , eachagent must receive at least two approved goods. Along with connectedness, this means that either S or S is not assigned to a s under A . Since the special goods are not approved by any other agent, wecan modify A to obtain another allocation, say A ′ , that is identical to A except for the allocation of thespecial goods, which are all assigned to the special agent. It is easy to see that A ′ is connected, EQ1, andhas egalitarian welfare at least . By an identical argument as in part (a), we can now infer a satisfying LNES assignment.We now move on to the proof of part (b) of Theorem 2, followed by that of part (d) which uses asimilar construction.
Proof. (of part (b)) We will once again show a reduction from
Linear Near-Exact Satisfiability ( LNES ). Construction of the reduced instance.
Let φ be an instance of LNES . We will begin with thedescription of the reduced instance.
Goods:
For every i ∈ [ p ] , we introduce one core good for every core clause denoted by U i , V i , U ′ i , V ′ i , and two auxiliary goods for every auxiliary clause denoted by C Li , C Ri . Next, we introduce twogoods for each shadow variable, i.e., corresponding to each of p i , q i , r i , s i , we introduce the shadow goods p i , p i , r i , r i , q i , q i , s i , s i . Finally, we introduce p dummy goods D , D , D , . . . , D p , D p , D p and two separator goods S , S . Thus, the total number of goods is m = 4 p +2 p +8 p +3 p +2 = 17 p +2 .The goods are arranged as shown in Figure 4. Agents:
As before, we have the main agents of the positive and negative type for every main vari-able x i , denoted by a x i and a ¯ x i , respectively. For every i ∈ [ p ] , the agent a x i approves the goods U i , V i , D i , D i , D i , while the agent a ¯ x i approves the goods U ′ i , V ′ i , D i , D i , D i . We also introduce a shadow agent for every shadow variable. If p i is a shadow variable occurring in core clause U i and aux-iliary clause C j , then the corresponding shadow agent p i approves the shadow goods p i , p i and theauxiliary goods C Lj , C Rj . The valuations of the other shadow agents r i , q i , s i are defined analogously.Lastly, we introduce a separator agent a that approves the two separator goods S , S . This completesthe construction of the reduced instance. Notice that the constructed instance is (5 , -sparse . Beforepresenting the proof of equivalence, we will prove a structural result in Lemma 2. Lemma 2.
In any
EF1 + PO allocation, the utility of the separator agent a is equal to . Moreover, forevery i ∈ [ p ] , exactly one of a x i or a ¯ x i is allocated the triplet of goods { D i , D i , D i } . roof. (of Lemma 2) Observe that in any EF1 and Pareto optimal allocation A , the separator goods S , S must be allocated to separator agent a . Indeed, S , S are valued only by a . If S , S areallocated to two distinct agents in some allocation A ′ , then A ′ can be shown to be Pareto dominated byanother allocation identical to A ′ except for the assignment of separator goods to the separator agent.Otherwise, if S , S are allocated to the same agent (different from a ) in A ′ , then EF1 is violated from a ’s perspective. Therefore, the utility of the separator agent a under any EF1 and Pareto optimalallocation is equal to . This implies that no main or shadow agent can obtain utility from goods inboth rows of Figure 4.To prove the second part of the lemma, we first observe that for every i ∈ [ p ] , the goods D i , D i , D i must be assigned between the main agents a x i and a ¯ x i in any Pareto optimal allocation. This is becausethese goods are approved only by a x i and a ¯ x i and no other agent. Furthermore, these agents can obtain autility of at most from the core goods. Therefore, any allocation A in which one or more of the dummygoods D i , D i , D i are assigned to agents other than a x i and a ¯ x i can be shown to be Pareto dominatedby another allocation, say A ′ , that is identical to A except for the assignment of these dummy goods,which are allocated exclusively among a x i and a ¯ x i .Next, suppose that both a x i and a ¯ x i are allocated only the dummy goods D i , D i , D i in a Paretooptimal allocation, say A . Assume, without loss of generality, that the utilities of a x i and a ¯ x i in A are and , respectively. Then, A can be shown to be Pareto dominated by another allocation thatis identical to A with the exception that one of the core goods, say U i , is assigned to a x i , and thetriplet { D i , D i , D i } to a ¯ x i , contradicting the Pareto optimality of A . Thus, the triplet of dummy goods { D i , D i , D i } must be completely assigned to either a x i or a ¯ x i . The Forward Direction.
Given a satisfying assignment τ for LNES , we will construct the desiredallocation as follows:• Allocate the separator goods S , S to the separator agent a .• If τ ( x i ) = 1 , then allocate { U i , p i , p i , r i , r i , V i } to agent a x i and { D i , D i , D i } to agent a ¯ x i . Inaddition, allocate { U ′ i , q i , q i } to q i , and { s i , s i , V ′ i } to s i . Recall that q i and s i are the shadowvariables that appear as negated literals in the core clauses U ′ i and V ′ i , respectively, along with ¯ x i .Otherwise, if τ ( x i ) = 0 , then allocate { U ′ i , q i , q i , s i , s i , V ′ i } to agent a ¯ x i and { D i , D i , D i } toagent a x i . In addition, allocate { U i , p i , p i } to p i , and { r i , r i , V i } to r i .• Finally, for every j ∈ [ p ] , allocate { C Lj } and { C Rj } to the two shadow agents whose correspond-ing literals satisfy the auxiliary clause C j .Notice that in the constructed allocation, each good is allocated to exactly one agent, and each agentreceives a connected interval. Also, the utility of the separator agent is , and exactly one agent corre-sponding to each variable receives a triplet of the corresponding dummy goods. Note that the utilityof each main agent is either or , and the utility of each shadow agent is either or . Furthermore,any main agent is allocated at most two goods valued by any shadow agent. Hence, the constructedallocation is EF1.We will now argue that the above allocation, say A , is Pareto optimal. Suppose for contradiction,that another allocation A ′ Pareto dominates A . The second part of Lemma 2 implies that in any EF1and Pareto optimal allocation, for every i ∈ [ p ] , the main agents a x i and a ¯ x i cannot both have utility . Thus, the utilities of the main agents under A and A ′ should be equal. Furthermore, one of themain agents corresponding to each variable will be allocated shadow goods corresponding to a pairof shadow agents (either { p i , r i } or { q i , s i } ). This implies that for at least one of these pairs, the twoshadow agents should each receive a utility of under A ′ . Hence, by a similar argument as above,21ll shadow agents will also have the same utility under A and A ′ , establishing that A ′ cannot Paretodominate A , as desired. The Reverse Direction.
We will now show a way to recover an
LNES assignment given a con-nected EF1 and Pareto optimal allocation, say A .Since A is EF1 and Pareto optimal, we know from Lemma 2 that the separator agent receives thetwo approved goods, and for each variable x i , exactly one of the corresponding main agents a x i or a ¯ x i receives the triplet of dummy goods { D i , D i , D i } . By EF1, the other main agent will achieve a utility by either receiving the interval { U i , p i , p i , r i , r i , V i } or { U ′ i , q i , q i , s i , s i , V ′ i } . This, in turn, forces at least one pair of shadow agents—either { p i , r i } or { q i , s i } —to obtain their utilities from the auxiliarygoods. Note that for any such pair, both agents will have a utility of at least due to EF1 condition.We will now show that exactly one of the two pairs of shadow agents derive their utility from theshadow goods, while the other pair meets the utility requirement though the auxiliary goods. Indeed,since there are p auxiliary goods (corresponding to p auxiliary clauses), at most p shadow agentscan obtain the desired utility from the auxiliary goods. Therefore, for every i ∈ [ p ] , exactly one pairof shadow agents—either { p i , r i } or { q i , s i } —are assigned shadow goods, while the other pair receivesauxiliary goods.Overall, we have that one set of p main agents gets exactly two core goods each (we will refer themas the “lucky” agents), while the other set of p main agents gets three dummy goods each (the “unlucky”agents). Notice that the two main agents corresponding to a main variable cannot both be lucky, norcan both be unlucky due to the argument presented in Lemma 2.This brings us to a natural way of deriving an LNES assignment τ from the allocation A . If themain agent of the positive (respectively, negative) type is unlucky, then we let τ ( x i ) = 0 (respectively, τ ( x i ) = 1 ). Furthermore, if A allocates a core good to a shadow agent, then the corresponding shadowvariable is set to , while shadow variables corresponding to shadow agents who receive auxiliary goodsare set to . Note that exactly p of the p shadow variables are set to under this assignment andthere are no conflicting assignments, implying that τ is indeed a valid solution to the LNES instance.This completes the proof of part (b) of Theorem 2.
Proof. (of part (d)) To prove part ( d ) , we adapt the construction in part (b) with a small change: For ev-ery i ∈ [ p ] , we introduce four auxiliary goods C L i , C L i , C R i , C R i instead of the original two C Li , C Ri .Note that such an instance is (6 , -sparse . We adapt the changes in the construction to the alloca-tion constructed in the forward direction by replacing C Li (respectively, C Ri ) with the set of goods { C L i , C L i } (respectively, { C R i , C R i } ). In the reverse direction, it is possible that under the givenallocation, say A , the main agents no longer receive all three dummy goods. Similar to the argumentin part (c), we can construct an allocation A ′ that is identical to A except that we allocate the triplet ofdummy goods { D i , D i , D i } to the corresponding main agent. At this stage, with a similar argumentas in the reverse direction of part (b), we can recover a satisfying LNES assignment.
Appendix B Proof of Theorem 4
Recall the statement of Theorem 4.
Theorem 4.
Given an instance with binary additive valuations and any agent ordering σ , a connected, σ -consistent, EQ1 , and PO * allocation of a path can be computed in polynomial time. We will start by describing the algorithm underlying this result, which, in turn, builds on Algo-rithm 1. This will be followed by a formal proof of Theorem 4.22 escription of the algorithm for EQ1 and PO* allocations:
Let σ := ( a , a , . . . , a n ) . Ouralgorithm for Theorem 4 consists of four phases. Phases 1 and Phase 2 are identical to those in Al-gorithm 1, and are used to find the optimal egalitarian welfare θ and the leftmost θ -unsafe agent a i ,respectively. Recall that in Phase 2, we also fix the allocations of the agents a , a , . . . , a i − .In the third phase, which we denote by Phase 3*, we partition the agents a i , . . . , a n in two groups as follows: We start with a partial allocation of the first i − agents a , a , . . . , a i − , and then considerthe remaining agents sequentially from left to right. That is, in round j ∈ { i, i + 1 , . . . , n } , we considerthe leftmost unallocated agent according to σ , namely a j . Starting with the leftmost available good, weallocate a minimal bundle worth θ + 1 to a j (note that θ + 1 is a realizable utility value under binaryvaluations). Next, the algorithm checks whether there exists a connected and σ -consistent allocationsuch that each subsequent agent receives utility θ (this step is similar to that in Algorithm 1). If thecheck passes (i.e., if there is a feasible partial allocation where the agents a j +1 , . . . , a n receive utility θ each), then we assign a j to group 1 and allocate to it the minimal bundle with utility θ + 1 (this is atemporary allocation). Otherwise, we assign a j to group 2 and allocate to it the minimal bundle worth θ . The above procedure is repeated for all subsequent agents, following which the algorithm proceedsto the fourth phase.In Phase 4*, we finalize the allocation of the agents a i , . . . , a n (recall that the allocation in Phase 3*is only tentative). At first, we mimic the allocation for agents a i , . . . , a n − from Phase 3*, and allocatethe remaining goods to a n . In this allocation, if u n ( A n ) ≤ θ + 1 , then the algorithm finalizes thebundles of all agents and returns the allocation. Otherwise, the algorithm performs a right-to-left scanof the path G similar to Phase 3 of Algorithm 1. In particular, starting from the rightmost availablegood, the algorithm moves leftwards along G and iteratively assigns minimal connected bundles withutility θ + 1 to group 1 or θ to group 2 agents in the reverse order a n , a n − , . . . , a i +1 . The remaininggoods are assigned to agent a i and the final allocation is returned as the output. Proof. (of Theorem 4) First, observe that the set of goods allocated in Phase 4* at least contains all thegoods allocated in a temporary allocation of Phase 3* (it may contain some additional goods). This isbecause, during a left-to-right partial temporary allocation in Phase 3*, we may not consider leftovergoods to the right of the allocated bundle for agent a n . Hence, in the final allocation in Phase 4*,algorithm has enough goods such that each group 1 agent receives a utility of at least θ + 1 , and eachgroup 2 agent receives a utility of at least θ .Next, we show that the allocation A returned by the algorithm is σ -consistent, complete, and EQ1.Notice that for the two cases in Phase 4*, in the last iteration, we allocate the leftover goods to agent a n or agent a i ; hence, completeness follows trivially. Also, A is σ -consistent because the algorithmmaintains this property at every step. Note that each of the agents a , . . . , a i − receives a bundle withutility θ + 1 under allocation A . In Phase 4*, consider the case when final allocation is a completion oftemporary partial allocation from Phase 3* by assigning leftover goods to agent a n . Here, it is easy tosee that the agents a i , . . . , a n are each allocated bundles with utility θ or θ + 1 . Hence, the allocation isEQ1. For the other case, when the final allocation is built with a right-to-left traversal and the leftovergoods are assigned to agent a i , it is easy to see that, except for agent a i all other agents receive a bundlewith utility either θ or θ + 1 . Moreover, a i belongs to group 2 (using the definition of θ -unsafe agent),and it receives a bundle with utility at least θ . Now, let S, S ′ be the set of goods allocated to agent a i under the allocation A , and allocation (say A ′ ) by Algorithm 1 for when we run it on the same instancerespectively. Observe that S ⊆ S ′ since the agents a i +1 , . . . , a n receive a minimal bundle with utilityeither θ or θ + 1 under the allocation A ′ while these agents receive a minimal bundle of utility exactly θ under the allocation A . At this stage, just like in the proof of Algorithm 1, we can conclude that u i ( S ) = θ . Hence, the allocation A is indeed EQ1.Finally, we show that A is PO* with a proof by contradiction. Let B be a σ -consistent EQ1 alloca-tion that Pareto dominates A . From Theorem 3, we know that the optimal egalitarian welfare for anyconnected and σ -consistent allocation is θ . Hence, each agent receives a bundle with utility either θ or23 + 1 under allocation A due to the way our algorithm works. Moreover, each agent receives a bundlewith utility either θ or θ + 1 under allocation B as θ is the optimal egalitarian welfare for the instanceand allocation B is EQ1. Let j be the leftmost agent such that u j ( A j ) < u j ( B j ) . We claim that j > i where a i is the θ -unsafe agent. This is because the agents a , a , . . . , a i − each receive a bundle withutility θ + 1 under allocation A and a i is the θ -unsafe agent. It is easy to see that u j ( B j ) = θ + 1 .Let A ′ be an allocation which is identical to allocation A for all agents a , a , . . . , a j − , and allocatesa minimal connected bundle to a j such that u j ( A ′ j ) = θ + 1 . Furthermore, let A ∗ (respectively, B ∗ ) bethe set of goods to the right of bundle A ′ j (respectively, B j ). We claim that B ∗ ⊆ A ∗ . This is becausefor all ℓ < j, u ℓ ( A ℓ ) = u ℓ ( B ℓ ) , and our algorithm allocated minimal bundles. But since u j ( A j ) = θ ,in Phase 3*, our algorithm labeled a j as group 1 agent. This implies that the set of goods A ∗ is notsufficient to ensure a utility θ for all subsequent agents. Since B ∗ ⊆ A ∗ , the allocation B is not EQ1which is a contradiction.Finally, let us turn to the running time analysis. Since we only consider binary valuations, the list L of all distinct realizable utility values contains at most m distinct values, and can be precomputed in O ( nm ) time. By a similar running time analysis as in the proof of Theorem 3, it follows that the totalrunning time for Phase 1 is O ( m ) , and that for Phase 2 is O ( nm ) .In Phase 3*, for each fixed j , in order to decide the group of agent a j , each of the unallocated goodsis considered towards at most one bundle. Hence, the total running time for this and each subsequentiteration is O ( m ) . Since there are at most n iterations, the total running time for Phase 3* is O ( nm ) .In Phase 4*, we finalize the allocation of the agents a i , . . . , a n by constructing at most two completeallocations corresponding to the two cases. Each good is considered towards at most one bundle in eachof these allocations. Thus, the algorithm requires O ( m ) time in Phase 4*. Hence, the overall runningtime of our algorithm is O ( m + nm ) .We close this section by noting that the problem of computing PO* allocations remains an interest-ing open question for general monotone valuations. Our algorithm for this problem does not extend toofar beyond the binary regime, as the following example shows: Consider an instance with four goods v , v , v , v and two agents a , a with valuations u = (1 , , , , u = (0 , , , . Suppose theagent ordering is σ = (1 , . The optimal egalitarian welfare is θ = 2 , and a is the leftmost θ -unsafe agent. On this instance, our algorithm returns the EQ1 allocation ( { v , v } , { v , v } ) , which is Paretodominated by the EQ1 allocation ( { v , v , v } , { v } ))