District-Fair Participatory Budgeting
D Ellis Hershkowitz, Anson Kahng, Dominik Peters, Ariel D. Procaccia
aa r X i v : . [ c s . G T ] F e b District-Fair Participatory Budgeting
D. Ellis Hershkowitz, Anson Kahng, Dominik Peters, Ariel D. Procaccia Carnegie Mellon University Harvard [email protected], [email protected], [email protected], [email protected]
Abstract
Participatory budgeting is a method used by city governmentsto select public projects to fund based on residents’ votes.Many cities use participatory budgeting at a district level.Typically, a budget is divided among districts proportionallyto their population, and each district holds an election overlocal projects and then uses its budget to fund the projectsmost preferred by its voters. However, district-level participa-tory budgeting can yield poor social welfare because it doesnot necessarily fund projects supported across multiple dis-tricts. On the other hand, decision making that only takesglobal social welfare into account can be unfair to districts:A social-welfare-maximizing solution might not fund any ofthe projects preferred by a district, despite the fact that its con-stituents pay taxes to the city. Thus, we study how to fairlymaximize social welfare in a participatory budgeting settingwith a single city-wide election. We propose a notion of fair-ness that guarantees each district at least as much welfare asit would have received in a district-level election. We showthat, although optimizing social welfare subject to this no-tion of fairness is NP-hard, we can efficiently construct a lot-tery over welfare-optimal outcomes that is fair in expectation.Moreover, we show that, when we are allowed to slightly re-lax fairness, we can efficiently compute a fair solution that iswelfare-maximizing, but which may overspend the budget.
Participatory budgeting is a democratic approach to theallocation of public funds. In the participatory budgetingparadigm, city governments fund public projects basedon constituents’ votes. In contrast to budget committees,which operate behind closed doors, participatory budgetingpromises to directly take the voices of the community intoaccount. Since 2014, Paris has allocated more than e Having separate elections for each district leads to severalproblems. Foremost, projects that are not local to a singledistrict cannot be accommodated. For this reason, Paris mustrun an additional election for city-wide projects. However,this splits the available budget for participatory budgetingbetween district-level and city-wide elections in an ad hocmanner, which is not informed by votes. Further, peoplemay have interests in multiple districts, such as those wholive and work in different districts. For this reason, Paris hasto allow residents to choose the district in which they vote.Lastly, a project that only benefits voters at the edge of a dis-trict may receive a number of votes that is not proportionalto the number of potential beneficiaries.A simple solution to these problems is a single city-wideelection. However, such a voting scheme may result in unfairoutcomes. For instance, if votes are aggregated to maximizesocial welfare (i.e., as is presently done in Paris on the dis-trict level) then it is possible that some districts might havenone of their preferred projects funded despite deservinga large proportion of the budget. Such outcomes are likelywhen some districts are much more populous than others, inwhich case projects local to small districts cannot gather suf-ficiently many votes. Ideally, we would like a system thatbalances the tradeoff between social welfare and fairnesswithout an arbitrary, pre-determined split between district-specific and city-wide funding. This motivates our centralresearch question:
How can we maximize social welfare in a way that isfair to all districts?
Intuitively, a solution that is fair to all districts shouldsomehow represent each districts’ constituents. One way toformalize this intuition is to stipulate that no district shouldbe able to obtain higher utility by purchasing projects withits proportional share of the budget. In particular, each dis- More specifically, projects are selected in descending order ofvote count until the budget runs out. In 2016, this split in Paris was e e
30 million for city-wide elections (Cabannes 2017). rict should receive at least as much utility as it would havereceived had it held a district-level election with its propor-tional share of the budget. We call this guarantee districtfairness . A district-fair allocation of funds always exists,since an outcome obtained by holding separate district elec-tions is district fair. We aim to find district-fair outcomes thatmaximize social welfare. Such an outcome will be a Pareto-improvement on the status quo of district-level participatorybudgeting, in the sense that each district’s welfare has in-creased.
Our Results.
In our model we think of (utilitarian) socialwelfare as induced by a given value assigned by each districtto each project; our goal is to maximize the sum of these val-ues over districts and selected projects. Note that this modelcaptures the setting of approval votes, where each voter de-cides on a collection of projects to vote for; the social wel-fare of a district for a project would then be interpreted asthe project’s overall number of approvals from voters in thatdistrict. This observation is important because some variantof approval voting is used in most real-world participatorybudgeting elections, including in Paris.We also assume that each district is endowed with an ar-bitrary fraction of the total budget. Clearly this captures, asa special case, the common setting where the endowment ofeach district is proportional to its size. Moreover, the reason-ing behind the existence of district-fair outcomes immedi-ately applies to the more general setting.We first show that it is NP-complete to compute an al-location that is welfare-maximizing subject to district fair-ness. This result holds even for the case of approval votesand proportional budgets, and therefore the generality ofour model only strengthens our positive (algorithmic) resultswithout weakening the main negative (hardness) result. Wealso show that the natural linear program (LP) formulationof the problem has an unbounded integrality gap. Since par-ticipatory budgeting elections can be large — hundreds ofprojects are proposed and hundreds of thousands of votesare cast in Paris — computational complexity can become aproblem in practice. Thus, we seek polynomial-time solu-tions with reasonable approximation guarantees.There are several ways one might relax our problem ortrade-off between parameters in our problem. In this work,we design polynomial-time algorithms that work when werelax or approximate some of the following: (1) the achievedsocial welfare; (2) the spent budget; (3) the fairness of thesolution; and (4) the absence of randomization.We first relax (4) by considering distributions overoutcomes, a.k.a. “lotteries”. We show that using amultiplicative-weights-type algorithm, one can efficientlyfind a lottery that guarantees budget feasibility (ex post), op-timum social welfare (ex post), and district-fairness in ex-pectation up to an ε (ex ante). Since the fairness guaran-tee only holds in expectation, some districts may be under-served once the lottery is realized. However, since participa-tory budgeting typically happens repeatedly (e.g., annually), Our notion of district fairness can be thought of as a form of in-dividual rationality where every district is seen as an “individual.” such districts could be compensated in the next election, forexample by increasing their share of the budget in the nextyear.We next consider what sort of deterministic guaranteesare achievable. To this end, we show how to use techniquesfrom submodular optimization to find an outcome that is dis-trict fair “up to one project” and which achieves optimumsocial welfare with the caveat that the outcome may needto spend . more money than was originally budgeted.We also give a randomized algorithm with the same guaran-tees but which overshoots the budget by only a /e ( ≈ fraction with high probability. Additionally, as a corollary ofthese results, we give both deterministic and randomized al-gorithms that achieve weaker utility and fairness guaranteesbut do not overspend the available budget. Related Work.
The social choice literature on par-ticipatory budgeting has both studied the voting rulesused in practice, and designed original voting schemes.Goel et al. (2019) study knapsack voting, used for exam-ple in Madrid (Cabannes 2014), where voters cannot ap-prove more projects than fit into the budget constraint.Talmon and Faliszewski (2019) axiomatically study a vari-ety of approval-based rules that maximize social welfare,both greedy and optimal ones.The unit cost case (where all projects have the samecost) is best-studied, as multi-winner or committee elections(Faliszewski et al. 2017). For example, this setting modelsthe election of a parliament. A main focus of that literatureis the computational complexity of the winner determina-tion of various voting rules. More relevant for our purposesare fairness axioms used in this setting. The most promi-nent such axioms are variants of justified representation (Aziz et al. 2017). These axioms are formulated for approvalvotes, and require that arbitrary subgroups of the electorateneed to be represented in the outcome if they are cohesive ,in the sense that there are a sufficient number of projectsthat are approved by every member of the subgroup. Severalvoting rules are known to satisfy these conditions, includ-ing Phragm´en’s rule and Thiele’s Proportional Approval Vot-ing (Janson 2016; S´anchez-Fern´andez et al. 2017; Brill et al.2017; Aziz et al. 2018). By contrast, district-fairness givesguarantees to a specific selection of subgroups (i.e., disjointdistricts) but does not require these groups to be cohesive.A very strong fairness axiom that is sometimes dis-cussed in the context of committee elections and partic-ipatory budgeting is the core (Fain, Goel, and Munagala2016; Aziz et al. 2017; Fain, Munagala, and Shah 2018). Itinsists that every subgroup (or coalition ) must be repre-sented (in the sense that it should not be possible forthe subgroup to propose an alternative use of their pro-portional share of the budget that each group memberprefers to the chosen outcome), without a cohesivenessrequirement. For approval-based elections, it is a majoropen question whether there always exists a core out-come. For general additive utilities, there are instancewhere no core outcome exists (Fain, Munagala, and Shah2018), but several researchers have proved the existence2f approximations to the core (Jiang, Munagala, and Wang2020; Fain, Munagala, and Shah 2018; Cheng et al. 2019;Peters and Skowron 2020). A district-fair outcome is, in asense, in the core: no subgroup which coincides with a dis-trict can block the outcome. Thus, our work shows that forgeneral utilities, a core-like outcome exists if we only allowa specific collection of (disjoint) coalitions to block.The problem of knapsack sharing (Brown 1979) has asimilar motivation to our problem. The knapsack sharingproblem supposes that the projects are separated intodistricts (instead of, in our case, the voters), and eachproject comes with a cost and a value. The aim is tofind a budget-feasible set of projects that maximize theminimum total value of the projects in a district. Note thatin this formulation all districts are treated equally (thereis no weighting by district population) and that there isno notion of the value of a project to a specific district.The literature contains a variety of algorithms for solv-ing this NP-hard problem (e.g., Yamada and Futakawa1997; Yamada, Futakawa, and Kataoka 1998;Hifi, M’Halla, and Sadfi 2005; Fujimoto and Yamada2006). Formally, the setting we consider is as follows. We aregiven a budget b ∈ Z > . There are m possible projects P = { x , . . . , x m } with associated nonnegative costs c : P → Z > . We refer to a subset W ⊆ P as an outcome . Thecost of an outcome W is c ( W ) := P x j ∈ W c ( x j ) . We saythat a subset W is budget-feasible if c ( W ) b .There are k districts d , . . . , d k . The social welfare (or utility ) that project x j provides to district d i is sw i ( x j ) ∈ Z > . We assume that utilities are additive; i.e., the util-ity that an outcome W ⊆ P provides to district d i issw i ( W ) := P x j ∈ W sw i ( x j ) . Furthermore, the total socialwelfare of W ⊆ P is sw ( W ) := P i ∈ [ k ] sw i ( W ) .Throughout this work we assume that sw ( x j ) and c ( x j ) are both poly ( k, m ) for each j . (A function f is poly ( x, y ) ifthere exists a k > such that f = O (( xy ) k ) .) We can relaxthis assumption using well-known bucketing techniques atthe cost of an arbitrarily small ε in the guarantees of our algo-rithms. See the fully polynomial time approximation schemefor the knapsack problem (Chekuri and Khanna 2005) for anexample of this technique.To model the participatory budgeting setting, we assumethat each district deserves some portion of the budget and, inturn, deserves at least the utility it could achieve if it spentits budget on its most preferred projects. Specifically, eachdistrict d i deserves some budget b i > where P i b i = b .District d i deserves utility f i := sw i ( W i ) , where W i :=arg max W : c ( W ) b i sw i ( W ) is d i ’s favorite outcome costingat most b i . Definition 1 (District-Fair Outcome) . We say that an out-come W is district-fair (DF) if sw i ( W ) > f i for all i . Computing f i is precisely an instance of the knapsackproblem; by our assumption that utilities and costs are poly-nomially bounded, this knapsack instance is solvable in poly- nomial time (Chekuri and Khanna 2005). Thus, we will as-sume f i is known.Note that the outcome S i W i is both budget-feasible anddistrict-fair, so an outcome with both properties always ex-ists. Our goal is to find a budget-feasible and district-fair out-come W which maximizes social welfare sw ( W ) . We callour problem district-fair welfare maximization . Through-out this paper, we let W ∗ := arg max W sw ( W ) be someoptimal solution, where the argmax is taken over budget-feasible and district-fair solutions. Similarly, we let OPT := sw ( W ∗ ) .We consider two relaxations of district fairness. The firstrelaxation extends the concept to lotteries over outcomes.We require that each district only needs to be approximatelysatisfied in expectation. We give an efficient algorithm tocompute optimal district-fair lotteries in Section 4. Definition 2 ( ε -District-Fair Lottery) . Given ε > ,we say that a probability distribution W over outcomesof cost at most b is an ε -district-fair ( ε -DF) lottery if E W ∼W [ sw i ( W )] > f i − ε for every district d i . The second relaxation is district-fairness up to one good (DF1). Intuitively, an allocation is DF1 if each district wouldbe satisfied if one additional project was funded.
Definition 3 (DF1) . An outcome W is DF1 if for every d i , sw i ( W ) + max x j ∈ ( P\ W ) sw i ( x j ) > f i . DF1 is inspired by the well-studied notion of EF1 (envy-freeness up to one good) from the private goods setting(Budish 2011). This relaxation is mild, and unlike relax-ations that require district-fairness to hold on average overdistricts, it is a uniform relaxation which provides guaran-tees for all districts. We study DF1 outcomes in Section 5.
Our first result shows that the problem of optimizing socialwelfare subject to district-fairness is NP-hard even in the re-stricted setting of approval votes (i.e., voters provide binaryyes/no opinions over projects) and budgets proportional todistrict sizes. In fact, our problem remains NP-hard in thisrestricted setting even when each district contains only onevoter and projects have unit costs.We reduce from exact 3-cover (X3C), which is known tobe NP-hard (Garey and Johnson 1979). The idea of our re-duction is as follows. Given an instance of X3C, we define adistrict for each of the elements in the universe, and then adda large amount of dummy districts. We then define a projectfor each set in our problem instance which gives one utilityto the districts corresponding to the elements which it cov-ers. We also define a large set of dummy projects that areapproved by all dummy districts. We then ask whether thereexists a district-fair outcome that attains high social welfare.An optimal solution for our district-fair welfare maximiza-tion problem, then, will first try to solve the X3C instanceas efficiently as possible so that it can spend as much of itsbudget as possible on high-utility dummy projects. We for-malize this idea in the following proof.3 heorem 1.
It is NP-complete to decide, given an instanceof district-fair welfare maximization and an integer M ,whether there exists a budget-feasible and district-fair out-come W such that sw ( W ) > M . NP-hardness holds even inthe restricted setting of approval votes and budgets propor-tional to district sizes, and when each district contains onevoter and all projects have unit cost.Proof. The stated problem is trivially in NP. For NP-hardness we reduce from X3C. In an instance of X3C, weare given a universe U = { e , . . . , e n } and a collection { S , . . . , S m } of 3-element subsets of U . It is a “yes”-instance if there exists a selection S j , . . . , S j n such that S j ∪ · · · ∪ S j n = U .Given an instance of X3C, we construct an instance ofour problem as follows. Let M = 3 mn + 1 . We have n + M districts, D ∪ D ′ . Let D = { d , . . . , d n } , whereeach d i in D corresponds to element e i . Additionally, let D ′ = { d n +1 , . . . d n + M } , where each d i ∈ D ′ is a dummydistrict. We have m + 2 n + M projects, X ∪ X ′ . Let X = { x , . . . , x m } , where x j ∈ X corresponds to set S j , andlet X ′ = { x m +1 , . . . , x m +2 n + M } , where each x j ∈ M ′ isa dummy project. Utilities are as follows: every dummy dis-trict approves every dummy project, so sw i ( x j ) = 1 for each i > n + 1 and x j ∈ X ′ . Also, each non-dummy districtapproves of non-dummy sets to reflect the structure of theX3C instance: that is, for each i n we have sw i ( x j ) = 1 if x j ∈ X and e i ∈ S j . All other utilities are 0: that is,sw i ( x j ) = 0 for all other i and j . Each project has cost 1,and our budget is b = 3 n + M . We assume all districts con-tain 1 voter, so b i = 1 for every district d i . Clearly, f i = 1 for each i . We ask whether there exists a district fair commit-tee with social welfare at least n + (2 n + M ) M .If there exists a solution S j , . . . , S j n to the X3C instance,then W = { x j , . . . , x j n } ∪ X ′ is an outcome with cost n + (2 n + M ) = 3 n + M = b . Clearly, W is district-fair,and its social welfare is n + (2 n + M ) M , so this is a “yes”-instance for the district-fair welfare-maximization problem.Conversely suppose that there exists a district-fair budget-feasible outcome W with social welfare at least n + (2 n + M ) M . Note that all projects in X together give overall wel-fare at most mn < M . Thus, we must have X ′ ⊆ W sinceotherwise the total welfare of W is less than (2 n + M ) M .Hence | X ∩ W | n . By district-fairness, for each i =1 , . . . , n , there must be some x j ∈ W such that e i ∈ S j .These two facts together imply that { S j : x j ∈ W } is asolution to the X3C instance.This NP-hardness result holds even if each district con-sists of a single voter and all projects have unit cost. As weshow in Appendix A in the supplementary material, this spe-cial case admits a polynomial-time -approximation. Our al-gorithm is based on a greedy algorithm and a combinatorialargument which “matches away” high utility goods of theoptimal solution. One might hope to achieve an approxima-tion result for the general case. A natural approach would beto round the optimal solution to the LP relaxation of the nat-ural ILP formulation of our problem. However, a simple ex-ample in Appendix B in the supplementary material shows that the integrality gap of that formulation is unboundedlylarge, so this approach will not work. In this section, we allow randomness and consider lotteriesover outcomes. Our main result for the lottery setting is an ε -DF lottery which always achieves the optimal social welfaresubject to district fairness. The welfare guarantee is ex post,so that every outcome in the lottery’s support achieves opti-mal welfare. For the remainder of this section we let ε > refer to the ε in the ε -DF definition. Theorem 2.
There is an algorithm which, in poly (cid:0) m, k, ε (cid:1) time, returns an ε -DF lottery W such that for all outcomes W in the support of W , we have sw ( W ) > OPT . The intuition for our algorithm is as follows. We begin byshowing that our problem is polynomial-time solvable if thenumber of districts k is constant. Such an algorithm is use-ful because we can artificially make the number of districtsconstant by convexly combining all districts into a singledistrict ˜ d . We can, then, compute as our solution a utility-optimal outcome W which is fair for ˜ d but not necessarilyfair for each d i individually. However, we can bias our solu-tion to try and satisfy fairness for certain districts by increas-ing the weights of these districts in our convex combination.Thus, if W is not fair for d i , we might naturally increasethe proportional share of d i in the convex combination andrecompute W in the hopes that the new outcome we com-pute will be fair for d i . We obtain our lottery by repeatedlyincreasing the weight of districts that do not have their fair-ness constraint satisfied, and then take a uniform distributionover the resulting outcomes.Turning to the proof, we begin by describing howto solve our problem in polynomial time when k is aconstant. Our algorithm will solve the natural dynamicprogram (DP). Specifically, consider the true/false value R ( sw (1) , . . . , sw ( k ) , j, b ) which is the answer to the ques-tion, “Does there exists an outcome of cost at most b us-ing projects x , x , . . . , x j wherein district d i achieves so-cial welfare at least sw ( i ) ?” If the answer to this questionis yes, then either the desired utilities are possible with thestated budget without using x j or there is an outcome whichuses at most b − c ( x j ) budget that doesn’t use x j in whichevery district gets at least its specified utility minus howmuch it values x j . Thus, R ( sw (1) , . . . , sw ( k ) , j, b ) is trueif and only if either R ( sw (1) , . . . , sw ( k ) , j − , b ) is true or R ( sw (1) − sw ( x j ) , . . . , sw ( k ) − sw k ( x j ) , j − , b − c ( x j )) is true, giving us a definition by recurrence.By our assumption that all costs and utilities are polyno-mially bounded, we can easily solve the dynamic program(DP) for the above recurrence, giving the following result. Lemma 3.
There is an algorithm that finds a budget-feasibledistrict-fair outcome W with sw ( W ) = OPT in m O ( k ) time.Proof. Our algorithm simply fills in the DP table and re-turns the outcome corresponding to the entry in our DP ta-ble which is true, satisfies sw ( i ) > f i for all i and which4aximizes P i sw ( i ) . The recurrence is correct by the abovereasoning.To see why we can fill in the DP table in the statedtime, note that we can trivially solve our base case, R ( sw (1) , . . . , sw ( k ) , j, , for each j and possible value foreach sw ( i ) in polynomial time. Since max i,j sw i ( x j ) is poly-nomially bounded in m , we need only check polynomially-many in m values for each sw ( i ) . Lastly, since j and b arebounded by a polynomial in m , we conclude that our DPtable has m O ( k ) entries, giving the desired runtime.We now describe our multiplicative-weights-type algo-rithm to produce our lottery using the above algorithm. Welet w ( t ) i > be the “weight” of district i in iteration t and let w ( t ) := P i w ( t ) i be the total weight in iteration t . Initially,our weights are uniform: w (1) i = 1 for all i .For any iteration t and district d i we let p ( t ) i := w ( t ) i w ( t ) bethe proportion of the weight that district i has in iteration t .These p ( t ) i will induce our convex combination over districts;in particular we let ˜ d ( t ) be a district which values project x j to extent ˜ sw ( t ) ( x j ) := P i p ( t ) i · sw i ( x j ) and which deserves ˜ f ( t ) := P i p ( t ) i · f i utility. Also, let sw max be the maximumwelfare of an outcome.With the above notation in hand, we can give our instanti-ation of multiplicative weights where T := kε · sw isthe number of iterations of our algorithm.1. For all iterations t ∈ [ T ] :(a) Let W t be an outcome that maximizes sw ( W t ) subjectto ˜ sw ( t ) ( W t ) > ˜ f ( t ) and c ( W t ) b . We can compute W t using Lemma 3.(b) Let m ( t ) i := sw i ( W t ) − f i be our “mistakes”, indicatinghow far off a district was from getting what it deserved.(c) Update weights: w ( t +1) i ← w ( t ) i · exp( − εm ( t ) i ) .2. Return lottery W , the uniform distribution over { W t } t .We now restate the usual multiplicative weights guaran-tee in terms of our algorithm. This lemma guarantees that,on average, the multiplicative weights strategy is competi-tive with the best “expert.” In the following h p ( t ) , m ( t ) i := P i p ( t ) i · m ( t ) i is the usual inner product. Lemma 4 (Arora, Hazan, and Kale 2012) . For all i we have T X t T h p ( t ) , m ( t ) i ε + 1 T X t T m ( t ) i . We can use this lemma to show the desired guarantees.
Proof of Theorem 2.
We use the algorithm described above.Our algorithm is polynomial time since it runs forpolynomially-many iterations and in each iteration we com-pute a solution for a problem on only one district which is We will only need to invoke the above algorithm for the case k = 1 . This amounts to solving the knapsack problem with a sin-gle covering constraint, which to our knowledge is not one of thestandard variants of the knapsack problem. solvable in polynomial time by Lemma 3. Also, note that byLemma 3 we know that c ( W t ) b for all t , so all outcomesin the lottery are budget-feasible.We now argue that the above lottery is utility-optimal. Fixan iteration t . Notice that since W ∗ is fair for all districtsthen it is fair for ˜ d ( t ) . In particular, ˜ sw ( t ) ( W ∗ ) = X i p i · sw i ( W ∗ ) > X i p i f i = ˜ f ( t ) Thus, W ∗ is a budget-feasible solution for the problem offinding a max-utility outcome which is fair for ˜ d ( t ) . Thus,sw ( W t ) can only be larger than sw ( W ∗ ) , meaning thatsw ( W t ) > OPT.We now argue that the above lottery is ε -DF in expecta-tion. Fix a district d i . By Lemma 4 we know that T X t T h p ( t ) , m ( t ) i ε + 1 T X t T m ( t ) i . (1)Now notice that by definition of m ( t ) i and since our lotteryis uniform over all W t we know that the right-hand-side ofEquation (1) is ε + 1 T X t T m ( t ) i = ε + 1 T X t ( sw i ( W t ) − f i )= ε − f i + 1 T X t sw i ( W t )= ε − f i + E W ∼W [ sw i ( W )] Thus, to show that f i − ε E W ∼W [ sw i ( W )] , it sufficesto show that the left-hand side of Equation (1) is at least .That is, we must show T P t T h p ( t ) , m ( t ) i . However,this amounts to simply showing that W t is fair for ˜ d ( t ) ; inparticular, we have that the left-hand-side is T X t T h p ( t ) , m ( t ) i = 1 T X t T X i p ( t ) i · ( sw i ( W t ) − f i )= 1 T X t T ˜ sw ( t ) ( W t ) − ˜ f ( t ) . It holds that ˜ sw ( t ) ( W t ) − ˜ f ( t ) > since we always choosea solution which is fair for ˜ d ( t ) , and so we conclude that theleft-hand-side of Equation (1) is at least . We now study how well we can do if we allow ourselvesto overspend the available budget. Certainly it is possibleto achieve district fairness and optimal fairness-constrainedutility OPT if the algorithm can spend double the availablebudget: we can compute an outcome W with c ( W ) b that is welfare-maximizing without attempting to satisfydistrict-fairness, and we can compute some outcome W with c ( W ) b that is district-fair (see Section 2); then W ∪ W satisfies district fairness and we clearly have c ( W ∪ W ) b and sw ( W ∪ W ) > OPT. In this section,we show that we can find a solution that requires less than5wice the budget, if we slightly relax the district fairness re-quirement to DF1. Our main result for the DF1 setting showsthat, under DF1 fairness, there is a deterministic algorithmwhich achieves DF1 and optimal social welfare if one over-spends a . fraction of the budget. Theorem 5.
For any constant ε > , there is a poly ( m, k ) -time algorithm which, given an instance of district-fair wel-fare maximization, returns an outcome W such that W isDF1, c ( w ) (1 .
647 + ε ) b , and sw ( W ) > (1 − ε ) OPT . Overspending by 64.7% is a worst-case result, and the al-gorithm may often overspend less. If the context does notpermit any overspending, one can run the same algorithmwith a reduced budget; then the output will be feasible forthe true budget, yet will satisfy weaker fairness and socialwelfare guarantees. More precisely, given an instance I anda multiplier β < , we define an instance I ′ ( β ) , which isidentical to I but in which each district d i contributes only β · b i and thus deserves utility f ′ i := sw i ( W ′ i ) , where W ′ i is d i ’s favorite outcome which costs at most β · b i . Addition-ally, let OPT ′ ( β ) represent the maximum achievable socialwelfare over all district-fair solutions in I ′ using a budget ofat most b ′ := β · b . Then, applying Theorem 5 to I ′ ( β ) re-sults in an outcome which is DF1 and utility-optimal on thisreduced instance and does not overspend the original budget b . Corollary 6.
For any constant ε > , there is a poly ( m, k ) -time algorithm which, given an instance I of district-fairwelfare maximization, returns an outcome W such that W is DF1 for I ′ ( . ) , c ( W ) (1 + ε ) b , and sw ( W ) > (1 − ε ) OPT ′ ( . ) . Our result uses a submodular optimization as a subroutine.If one allows randomization in this subroutine, algorithmswith better approximation ratios are known. Thus, we canprove a similar theorem (and corollary) with a randomized algorithm which achieves DF1 and optimal social welfarewhile overspending its budget by only a e ≈ . fractionof the budget, with high probability (i.e., with probability − p ( m,k ) where p ( m, k ) is some polynomial in m and k ).We defer details of our randomized algorithm to Appendix Cin the supplementary material.In the remainder of this section, we will prove Theorem 5.Our main tool is a notion of the “coverage” of a partial out-come. An outcome has high coverage if we do not need tospend much more money to make it district-fair. On a highlevel, our proof consists of two main steps. First, we showhow to complete an outcome with good coverage into a DF1outcome. Second, we will show how to frame the problemof finding a solution with good coverage and social welfareas a submodular maximization problem subject to linear con-straints, allowing us to use a result by Mizrachi et al. (2018).We begin by formalizing the coverage of a solution.Roughly, if we imagine that initially every district requiresits portion of the budget for fairness, then fractional cover-age captures how much less districts must spend to satisfytheir own fairness constraints. Thus, if we imagine that ouralgorithm first spends its budget to satisfy fairness as effi-ciently as possible, and then spends the remainder of its budget on the highest utility projects, then the coverage ofa collection of projects is roughly how much budget this col-lection “frees up” for the algorithm to spend on the highestutility projects. More formally, we define coverage by wayof the notions of fractional outcomes and residual budget re-quirements. Definition 4 (fractional outcomes) . A fractional outcomeis a vector p ∈ R m where p j . We overloadnotation and let the social welfare of p for district d i be sw i ( p ) := P j sw i ( x j ) · p j . Similarly the social welfare of p is P i sw i ( p ) . Lastly, we define the cost of p as P j c ( x j ) · p j . We now define the residual budget requirement of a dis-trict, given an outcome, which can be understood as the min-imum amount of additional money that must be spent to sat-isfy the district, if fractional outcomes are allowed.
Definition 5 (resid i ( W ) ) . The residual budget requirementof district d i given (integral) outcome W is the minimumcost of a fractional outcome p such that sw i ( W ) + sw i ( p ) > f i and p j = 0 for all x j ∈ W . We can now define the coverage of an outcome for a par-ticular district i in terms of the total amount of budget theydeserve and their residual budget requirement. Definition 6 (cover i ( W ) ) . The coverage of an outcome W for district d i is the difference between the amount of bud-get they deserve, b i , and their residual budget requirement: cover i ( W ) := b i − resid i ( W ) . Lastly, we define the coverage of an outcome.
Definition 7 (cover ( W ) ) . The overall coverage of an out-come W is the sum over all districts d i of the coverage W affords d i : cover i ( W ) := P i cover i ( W ) . Next, we establish a useful property of DF1 solutions. Inparticular, given a set of projects that achieves relativelygood fairness on average , we can then buy a small subsetof projects that results in fairness up to one good for all dis-tricts. In particular, given a collection of projects that coversa − β fraction of all fairness constraints, we can use at mostan extra β fraction of our budget in order to complete this toa DF1 solution. Moreover, this completion is quite intuitive:purchase all projects whose total coverage exceed their cost,until there are no such projects remaining.Formally, we state the following DF1 completion lemma.
Lemma 7 (DF1 Completion) . Given an outcome W with cover ( W ) = b − r , one can compute in polynomial time aset W ′ ⊇ W such that W ′ is DF1 and c ( W ′ ) c ( W ) + r .Proof. We first prove that for every non-DF1 outcome W ,there exists a project that we can add to W which increasesits coverage by at least c ( x j ) . Suppose that W is an outcomethat fails DF1, and let d i be a district such that sw i ( W ) + sw i ( x j ) < f i for all x j W . Let p be the fractional out-come witnessing resid i ( W ) ; thus sw i ( W ) + sw i ( p ) > f i .We may assume without loss of generality that all but at mostone project is integral in p j (because there is always someoptimal p with this property by additivity of sw i ). Since W fails DF1 for d i , there is some x j W such that p ( x j ) = 1 .Then resid i ( W ∪ { x j } ) = resid i ( W ) − c ( x j ) (witnessed by6he fractional outcome obtained from p by removing x j fromit). Thus, from definitions, cover i ( W ∪{ x j } ) = cover i ( W )+ c ( x j ) , and hence cover ( W ∪ { x j } ) > cover ( W ) + c ( x j ) .Now suppose we are given an outcome W withcover ( W ) = b − r , which fails DF1. We can identify aproject x j as above, add it to W , and increase the cover-age by at least c ( x j ) . We repeat this until the outcome isDF1. This process must stop, since at each step the cover-age increases by c ( x j ) but by definition the coverage cannever exceed b . For the same reason, the cost of the projectswe have added to W cannot exceed r , and thus c ( W ′ ) c ( W ) + r .With this lemma in hand, we now turn to the problemof finding high-coverage outcomes with good welfare. Let B > be a lower bound on the social welfare we desire. Werephrase our problem as an optimization problem in whichwe maximize the coverage of an outcome subject to a lin-ear knapsack constraint and a linear covering constraint. Theknapsack constraint enforces budget feasibility, and the cov-ering constraint encodes the requirement that the total utilityof the outcome is at least B . max W ⊆P cover ( W ) s.t. sw ( W ) > B,c ( W ) b. (DF1P)The main tool we apply is a theorem on the maximizationof nondecreasing submodular functions of Mizrachi et al.(2018). Recall that a set function is nondecreasing if its valuenever decreases as elements are added to its input, and sub-modular if it exhibits diminishing returns. Definition 8.
Given a finite set Ω , a set function f : Ω → R > is nondecreasing and submodular if for every A, B ⊆ Ω such that A ⊆ B we have f ( A ) f ( B ) and f ( A ∪ { x } ) − f ( A ) > f ( B ∪ { x } ) − f ( B ) for all x ∈ Ω \ B . The theorem we apply is as follows.
Theorem 8 (Mizrachi et al. 2018, Theorem 5) . For eachconstant ε > , there exists a deterministic algorithm formaximizing a nondecreasing submodular function subjectto one packing constraint and one covering constraint thatruns in time O ( n O (1) ) , where n = | Ω | is the size of the sup-port of the set function, satisfies the covering constraint upto a factor of − ε and the packing constraint up to a factorof ε , and achieves an approximation ratio of . . We apply this theorem to find a solution that satisfies a . fraction of coverage and achieves optimal fairness-constrained utility. Then, we apply Lemma 7 to augment oursolution using an additional − . ε fraction of our bud-get in order to obtain a final solution which satisfies full DF1.However, in order to apply Theorem 8, we must first estab-lish that cover ( W ) is a nondecreasing submodular function.In particular, note that the coverage functions cover i ( W ) foreach district are clearly nondecreasing and submodular. Itfollows that their sum, cover ( W ) is also nondecreasing andsubmodular, yielding the following lemma. Lemma 9.
The function cover ( W ) is nondecreasing andsubmodular. We are now ready to prove Theorem 5, which applies theDF1 completion lemma to an approximately optimal solu-tion for the problem DF1P.
Proof of Theorem 5.
Recall that we have assumed that themaximum utility of an outcome is polynomially bounded in m and k and that the maximum utility is integral. Thus, thevalue of OPT falls in a polynomial range. For each value B in this range, solve the problem DF1P using the algorithmfrom Theorem 8. Now consider all values of B for whichthe algorithm returned a solution with cover ( W ) > . b ;such a value must exist since we are guaranteed this con-dition when B = OPT (since for this value, the optimumof problem (DF1P) is b ). Among all solutions we foundthat satisfy cover ( W ) > . b , take the one that maxi-mizes sw ( W ) . This solution provides social welfare at least (1 − ε ) OPT.We have obtained an outcome W withcover ( W ) > . b = b − . b, and sw ( W ) > (1 − ε ) OPT and c ( W ) (1+ ε ) b . Now applyLemma 7 to W to obtain a DF1 outcome W ′ ⊇ W with c ( W ′ ) c ( W ) + 0 . b (1 + 0 .
647 + ε ) b. This outcome W ′ satisfies the requirements of Theorem 5. Our results extend to the special case of unit costs, alsoknown as committee selection. In committee selection, weelect a committee to represent voters in a larger governmen-tal body such as a parliament. Often, to ensure local repre-sentation, the electorate is split into voting districts, whichelect their representatives separately. The districts may beapportioned different numbers of representatives, for exam-ple based on district size. While this scheme guarantees eachdistrict representation, it may well be possible to increasethe welfare of the voters in a district, for example by elect-ing a diverse array of candidates with expertise in variousareas who can gather votes from across the electorate. Thus,it is natural for all districts to elect the committee togetherif we impose district-fairness constraints. This way, we canmaximize social welfare of the final committee while guar-anteeing each district fair representation. This gives a moreholistic view of committee selection in exactly the same waywe addressed participatory budgeting, only instead of pool-ing the budget between districts, we now pool seats on acommittee.Our model implicitly treats districts as atoms, and so dis-trict fairness is a kind of individual rationality property. Inturn, individual rationality is a type of strategyproofness: itincentivizes districts not to leave the central election andinstead hold a separate one. Is it possible to design a vot-ing scheme that is fully strategyproof for districts, so thatdistricts do not have incentives to misreport the utilities oftheir residents? Unfortunately not: Peters (2018) proves animpossibility theorem about committee elections which im-plies that there does not exist a voting rule that is efficient,7istrict-fair, and also strategyproof. This result holds evenfor approval votes.Several open questions remain. Most obvious is the ques-tion of whether can we achieve welfare maximization andDF1 in polynomial time while guaranteeing to overspendthe budget by less than /e . More broadly, it would be inter-esting to study our problem with more general utility func-tions such as submodular or even general monotone valua-tion functions. Additionally, it would be exciting to study ap-proximation algorithms which promise full district fairness.In Appendix B in the supplementary material, we presentan algorithm which satisfies district fairness and provides a / -approximation to optimal district-fair social welfare inthe special case of unanimous districts; it would be interest-ing to extend this result to the general case. References
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EuropeanJournal of Operational Research A -Approximation for Unanimous Districtswith Unit Costs In this section we study approximation algorithms for thesimplest version of our problem which we know to be NP-hard: when each districts consist of a single voter and ev-ery project has unit cost. In fact, we will study a strictlymore general setting than each district consisting of a sin-gle voter; namely, we study the setting where each districtis “unanimous.” Formally, we study instances of district-fair welfare maximization where c ( x j ) = 1 for all j andsw i ( x j ) ∈ { , | d i |} for all i where | d i | is the number of vot-ers in d i . For this setting we will give a -approximation.Our algorithm will make use of the following notion ofconditional coverage which builds on Definition 6. Definition 9.
The coverage of a project x j given an outcome W is cover ( x j | W ) := P i cover i ( W ∪ { x j } ) − cover i ( W ) . Notice that by our assumption of unit cost and unanimousdistricts we have that cover ( x j | W ) ∈ Z > .We now present our greedy algorithm that satisfies dis-trict fairness and achieves a / -approximation to the opti-mal district-fair utility for the setting of unanimous districtsand unit costs. Formally, the algorithm, which we call the Unanimous Greedy Algorithm (UGA) proceeds as follows.1. Given an instance I , initialize W ← ∅ .2. For j ∈ [ b ] :(a) Let c j := max x j cover ( x j | W j − ) be the max possiblecoverage and let X j := { x j : cover ( x j | W j − ) = c j } be all projects which achieve this coverage.(b) Let x j := arg max x j ∈ X j sw ( x j ) be the max coveringproject with maximum utility.(c) Update W j ← W j − ∪ { x j } .3. Return W b . Theorem 10.
Given an instance I consisting of unanimousapproval districts, UGA returns a solution which satisfiesdistrict fairness and achieves a / -approximation to the op-timal district-fair utility.Proof. Let W represent the result of UGA, and let W ∗ represent the optimal district-fair outcome. Furthermore, let N , N and N be all projects purchased by UGA that hadconditional coverage at least 2, exactly 1 and exactly 0 when purchased by UGA respectively. Clearly W is district-fair and budget-feasible and so we need only argue that itachieves at least sw ( W ∗ ) / utility.Now, consider the following subproblem, which we willcall I ′ , which intuitively is our original instance I but whereall of N is forced to be in a solution and no projects from N are available. More formally, I ′ is I but where our budgetis changed to b ′ := b − | N | − | N | , f i is changed to f ′ i :=max( f i − sw i ( N ) , for all i and the set of purchasableprojects is P ′ := P \ N . sw i ( x j ) is the same for all i, j in I ′ as in I . Notice that the coverage of any project in I ′ is at most but the total coverage required for fairness is b ′ ,meaning that every budget-feasible and district-fair solutionfor I ′ has size exactly b ′ . Also notice that N is not onlyfeasible for I ′ but also attains the optimal utility among alldistrict-fair and budget-feasible solutions.We claim that there exists a subset N ∗ ⊆ W ∗ \ N whichis district-fair and budget-feasible for I ′ . To see this, notethat we can iteratively build N ∗ by initializing it to ∅ andthen repeatedly adding to it any x j ∈ W ∗ \ N ∗ \ N suchthat cover ( x j | N ∗ ) in I ′ is at least . After b ′ such additionswe are guaranteed to have a district-fair and budget-feasiblesolution for I ′ and such an x j always exists since W ∗ \ N is district-fair for I ′ . As noted above, any district-fair andbudget-feasible solution for I ′ has size b ′ and so | N ∗ | = b ′ .Thus, since N is optimal for I ′ , we know that N achieves at least as high utility as N ∗ , i.e., | N | = | N ∗ | = b ′ ,and sw ( N ∗ ) sw ( N ) . (2)It remains to understand the utility of W ∗ \ N ∗ . However,note that at least half of the projects other than N must bein the N phase. That is, | N | > | N | / . Intuitively, thismeans that UGA “frees up” at least b − b ′ money to spendon high-utility projects. Let P := P \ { N ∪ N } be allprojects not in N or N . We have that sw ( N ) is the utilityof the top b − b ′ projects in P . On the other hand, considerprojects in W ∗ \ N ∗ . We can divide these into projects whichare in N and N and those which are not. In particular, let W ∗ , := W ∗ \ N ∗ ∩ ( N ∪ N ) and let W ∗ := W ∗ \ N ∗ ∩P so that W ∗ , ∪ W ∗ = W ∗ \ N ∗ . Now notice that triviallysw ( W ∗ , ) sw ( N ) + sw ( N ) . (3)On the other hand, | W ∗ | = b − b ′ − | W ∗ , | b − b ′ and W ∗ ⊆ P and so sw ( W ∗ ) is at most the utility of the b − b ′ highest utility projects in P . Since our utilities are additiveand sw ( N ) is the utility of the b − b ′ highest utility projectsin P , it follows thatsw ( W ∗ ) · sw ( N ) . (4)Since W ∗ = N ∗ ∪ W ∗ ∪ W ∗ , , we can combine the abovebounds to conclude our -approximation. Namely, applyingthe additivity of our utilities and combining Equations 2, 3and 4 we havesw ( W ∗ ) = sw ( N ∗ ) + sw ( W ∗ ) + sw ( W ∗ , ) · sw ( N ) + 2 · sw ( N ) + sw ( N ) · sw ( W ) and so we conclude that sw ( W ) > sw ( W ∗ )2 . B Integrality Gap
Here, we investigate the integrality gap of the natural LP forour problem. As a reminder, the integrality gap of an LP mea-sures how much better a fractional solution can do than anintegral solution. An unbounded integrality gap shows thatany analysis of an approximation algorithm which chargesthe value of its integral solution to the value of the optimalLP gives an unboundedely-bad approximation ratio. For thisreason integrality gaps are sometimes taken as evidence ofhardness of approximation. For more details on the topic ofintegrality gaps see Williamson and Shmoys (2011).We will show that our LP has an unbounded integralitygap which suggests that approximation algorithms which re-turn budget-feasible and district-fair solutions with nearly-optimal social welfare may be difficult or impossible to at-tain for the general case.Formally our LP and its integrality gap are as follows. OurLP has a variable y j for each project x j corresponding to theextent to which we choose x j . max X j y j · sw ( x j ) s.t. X j y j · c ( x j ) b X j y j · sw i ( x j ) > f i ∀ i y j ∀ j (DFLP)We let DF LP ( I ) correspond to the polytope correspond-ing to the above LP for an instance I of district-fair welfaremaximization.The integrality gap of DFLP is defined as min I max y ∈ DF LP ( I ) ∩ Z m P j y j · sw ( x j )max y ∈ DF LP ( I ) P j y j · sw ( x j ) . The basic idea of our integrality gap construction is as fol-lows. We will construct an instance of social-welfare max-imization where the preferences of each district are “circu-lar”. In particular, each district will like two projects andevery project will be liked by exactly two districts. As inour NP-hardness proof, we will also have a collection ofdummy projects which are given very high utility by dummydistricts which deserve no utility. An optimal fractional so-lution will be able to choose each non-dummy project to ex-tent essentially to satisfy district-fairness and then spendits remaining budget on high-utility dummy projects. On theother hand, the optimal integral solution will have to spendits entire budget satisfying fairness. Theorem 11.
There does not exist a function f such thatthe integrality gap of DFLP is at most f ( k,m ) . Further, thisintegrality gap holds even when all projects have unit cost. Proof. Fix k ∈ Z > and a sufficiently small ε > .We define our instance of social-welfare maximization on k districts where d , d , . . . d k − will be non-dummy dis-tricts and the district d k will be a dummy district. Simi-larly, we will have k − projects where x , x , . . . , x k − will be non-dummy projects and the remaining projects x k , . . . , x k − will be dummy projects.For each non-dummy district d i we let b i = 1 and defineits utility for project x j assw i ( x j ) := ε if j = i if j = ( i + 1 mod k −
1) + 10 otherwiseFor the dummy district d k we let b k = 0 and define itsutility for project x j assw k ( x j ) := (cid:26) B if x j is a dummy project otherwisefor B sufficiently large to be chosen later. Notice thatsw ( x j ) = B for each dummy project x j . Lastly, we let ourbudget b = k − and we let c ( x j ) = 1 for all x j .Now notice that each non-dummy district d i has f i =1 + ε . Consequently, any district-fair integral solution mustinclude all non-dummy projects, namely x , x , . . . , x k − .However, since b = k − , it follows that the only district-fair integral solution is W int := { x , x , . . . , x k − } wheresw ( W int ) = (2 + ε )( k − .On the other hand, consider the following fractional solu-tion y . For each non-dummy project x j we let y j = ε .For each dummy project x j we let y j be (1 − ε ) . Clearly P j y j b . Moreover, notice that for each district d i wehave P j y j · sw i ( x j ) = ε (2 + ε ) > ε = f i andso our solution is indeed in the polytope of DFLP. However,since y j = (1 − ε ) for each dummy project we have that P j y j · sw ( x j ) > B ( k − − ε ) .Thus, for the above instance we have that the ratio of theoptimal integral solution to the optimal fractional solution isat most (2 + ε )( k − B ( k − (cid:0) − ε (cid:1) B .
Since B can be chosen independently of k and m , we havethat the above instance has integrality gap strictly less than f ( k,m ) for any function f of k and m .We note that the proof of the above result also rules outany integrality gap which is which is larger than o ( c ) where c is the total number of voters across all districts. C Randomized Optimal DF1 Outcome withExtra Budget
In this section we give our randomized analogues of The-orem 5 and Corollary 6. We use the notation of Section 5throughout this section. Whereas our deterministic algo-rithms overspend budget by . , our randomized algo-rithms will only overspend it by e with high probability. For-mally, we show the following theorem.10 heorem 12. There is a poly ( m, k ) -time algorithm which,given an instance of district-fair welfare maximization, re-turns an outcome W such that W is DF1, c ( W ) (cid:0) e + ε (cid:1) b ≈ . b with high probability, and sw ( W ) > (1 − ε ) OPT for any fixed constant ε > . As with Corollary 6 for Theorem 5, we immediately havea corollary which gives an algorithm which does not over-spend its budget.
Corollary 13.
There is a poly ( m, k ) -time algorithm which,given an instance of district-fair welfare maximization, re-turns an outcome W such that W is DF1 for I ′ (cid:16) /e (cid:17) , c ( W ) (1 + ε ) b with high probability, and sw ( W ) > (1 − ε ) OPT ′ (cid:16) /e (cid:17) for any fixed constant ε > . On a high level, this proof will closely follow that of The-orem 5. In particular, it uses the same submodular optimiza-tion framing of the problem (i.e., DF1P). However, thereare some notable differences. In particular, we leverage thefollowing randomized result from Mizrachi et al. (2018) in-stead of the previous deterministic result from Mizrachi et al.(2018).
Theorem 14 (Mizrachi et al. 2018, Theorem 1) . For eachconstant ε > , there exists a randomized algorithm for max-imizing a nondecreasing submodular function subject to onepacking constraint and one covering constraint that runs intime O ( | Ω | O (1) ) , satisfies the packing constraint, satisfiesthe covering constraint up to a factor of − ε , and achievesan expected approximation ratio of − e − ε . Additionally, we require Hoeffding’s inequal-ity (Hoeffding 1963), which bounds the probabilitythat the sum of a sequence of independent random variablesdeviates from its expectation; we restate it below.
Theorem 15 (Hoeffding’s Inequality) . Given a sequence of n independent random variables Y , . . . , Y n , where each Y i takes a value in the range [ α i , β i ] , we have that Pr E " n X k =1 Y k − n X k =1 Y k > t ! e − t P ni =1( βi − αi )2 , where t > . We now present the main lemma of the section, whichstates that there exists a polynomial-time algorithm which,given a guess of the optimal district-fair social welfare B OPT, returns an outcome W that is DF1 fair, overspendsthe budget by approximately b/e with high probability, andachieves at least (1 − ε ) B social welfare. Lemma 16.
Given an instance of DF1P, there is an algo-rithm that runs in polynomial time which returns an outcome W such that W is DF1, c ( W ) (cid:0) e + ε (cid:1) b with highprobability, and sw ( W ) > (1 − ε ) B , where ε > is anarbitrary constant and B OPT .Proof.
Note that, by Lemma 7, it suffices to find an outcome W ′ such that c ( W ′ ) b , cover ( W ′ ) > (cid:0) − e − ε (cid:1) b , andsw ( W ′ ) > (1 − ε ) B , as we can complete this solution intoa DF1 solution with (cid:0) e + ε (cid:1) b more budget. Let ε := ε/ . By Lemma 9, we may apply Theorem 14in order to find a solution W such that cover ( W ) > (1 − ε )(1 − /e ) in expectation, sw ( W ) > B , and c ( W ) b .We now show how to transform this guarantee in expectationinto one with high probability by using Hoeffding’s inequal-ity and an averaging argument.In order to apply Hoeffding’s inequality to our setting,let Y j represent the coverage of the j th run of our ap-plication of Theorem 14. We know that each run is inde-pendent, and therefore the Y j ’s are also independent. Let n = ω (log k/ε ) , and define a sequence of n independentrandom variables Y , . . . , Y n representing the coverage of n runs of the mechanism. Furthermore, because coverage isbounded between 0 and b , we have that β i = b and α i = 0 for all i ∈ [ n ] .By Hoeffding, we have Pr E n X j =1 Y j − n X j =1 Y j > ε (cid:18) − e (cid:19) bn exp − ε (cid:0) − e (cid:1) bn ) P ni =1 ( β i − α i ) ! = exp − ε (cid:18) − e (cid:19) n ! . Because we set n = ω (log k/ε ) , this probability goesto 0 polynomially quickly. Therefore, we know that, withhigh probability, E hP nj =1 Y j i − P nj =1 Y j ε (cid:0) − e (cid:1) bn ,or P nj =1 Y j > bn (1 − ε ) (cid:0) − e (cid:1) − ε (cid:0) − e (cid:1) bn . By anaveraging argument, this means that, with high probability,there exists a Y i such that Y i > (1 − ε − ε ) (cid:0) − e (cid:1) b =(1 − ε ) (cid:0) − e (cid:1) b , as desired.With this lemma in hand, we are ready to prove Theo-rem 12. Proof of Theorem 12.
Recall that we have assumed that themaximum utility of an outcome is polynomially bounded in m and k and that the maximum utility is integral. Thus, thevalue of OPT falls in a polynomial range. For each value B in this range, run the procedure from Lemma 16. By aunion bound over these polynomially-many applications ofLemma 16, we have that all applications of Lemma 16 with B OPT succeed with high probability, resulting in a col-lection of solutions; one for each B . Now consider all val-ues of B for which the algorithm returned a solution with c ( W ) (1+ e + ε ) b ; such a value must exist since with highprobability we are guaranteed this condition when B = OPTand we know that, with high probability, all applications ofLemma 16 succeed for all B OPT. Among all solutionswe found that satisfy c ( W ) (1 + e + ε ) b , take the one thatmaximizes sw ( W ) . This solution provides utility at leastat least