Estimating the Nash Social Welfare for coverage and other submodular valuations
aa r X i v : . [ c s . G T ] J a n Estimating the Nash Social Welfarefor coverage and other submodular valuations
Wenzheng Li ∗ Jan Vondr´ak † Abstract
We study the Nash Social Welfare problem: Given n agentswith valuation functions v i : 2 [ m ] → R + , partition [ m ] into S , . . . , S n so as to maximize ( Q ni =1 v i ( S i )) /n . The problemhas been shown to admit a constant-factor approximationfor additive, budget-additive, and piecewise linear concaveseparable valuations; the case of submodular valuations isopen.We provide a e (1 − e ) -approximation of the optimalvalue for several classes of submodular valuations: coverage,sums of matroid rank functions, and certain matching-basedvaluations. Nash Social Welfare is an optimization problem in thefollowing form:
Nash Social Welfare (NSW).
Given m indi-visible items and n agents with valuation functions v i : { , } m → R + , we want to allocate the items to theagents, that is find x ∈ { , } n × m such that for each j , P ni =1 x ij ≤
1, in order to maximize the geometricaverage of the valuations, ν ( x ) = n Y i =1 v i ( x i ) ! /n . The notion of Nash Social Welfare goes back toJohn Nash’s work [19] on bargaining in the 1950s. Itcan be viewed as a compromise between MaximumSocial Welfare (maximizing the summation P ni =1 v i ( x i ),which does not take fairness into account), and Max-Min Welfare (maximizing min ≤ i ≤ n v i ( x i ), which isfocusing on the least satisfied agent ignores the possibleadditional benefits to others). Nash Social Welfare alsocame up independently in the context of competitiveequilibria with equal incomes [20] and proportionalfairness in networking [15]. An interesting feature ofNash Social Welfare is that the optimum is invariantunder scaling of the valuations v i by independent factors λ i ; i.e., each agent can express their preference ina “different currency” and this does not affect theproblem. ∗ Stanford University † Stanford University
The difficulty of the problem naturally dependson what class of valuations v i we consider. Unlikethe (additive) Social Welfare Maximization problem,the Nash Social Welfare problem is non-trivial even inthe case where the v i ’s are additive , that is v i ( x i ) = P mj =1 w ij x ij . It is NP-hard even in the case of 2agents with identical additive valuations (by a reductionfrom the Subset Sum problem), and APX-hard formultiple agents [16]. Constant-factor approximationsfor the additive case were discovered only recently,in remarkable works by Cole and Gatskelis [7], andsubsequently via a very different algorithm by Anari etal. [2]. Inspired by the exciting breakthrough, a seriesof followup work have also been developed along thisline [3, 6].A natural question is whether a constant-factor ap-proximation extends to broader classes of valuations, forexample submodular valuations (where a constant fac-tor is known for Maximum Social Welfare [10,21]). Thisis unknown at the moment. Some progress has beenmade for classes of valuations beyond additive ones: aconstant factor for concave piece-wise linear separableutilities [1], and for budget-additive valuations [4, 11].The problem seems particularly challenging for gen-eral submodular valuations. Only very recently, an O ( n log n )-approximation has been designed for sub-modular valuations in [12]. (The authors also present a(1 − /e − ǫ )-approximation in the case where the numberof agents is constant, which follows from earlier work onmultiobjective submodular optimization.) We consider the Nash Social Wel-fare optimization problem for several subclasses of sub-modular valuations. Our main results are constant-factor approximations of the optimal value for severalsuch classes that we discuss below. We stress that wedo not know how to find an allocation of correspond-ing value in polynomial time, which is a situation rem-iniscent of certain variants of max-min allocation. Wediscuss this in more detail below.The classes of submodular valuations that we con-sider are: • Matroid rank functions: Given a matroid M = Copyright © [ m ] , I ), its rank function is r M ( S ) = max {| I | : I ⊆ S, I ∈ I} . More generally, for w ∈ R m + , a weighted matroidrank function is r M , w ( S ) = max { X j ∈ I w j : I ⊆ S, I ∈ I} . We obtain a e (1 − e ) -approximation for this class. • The cone generated by matroid rank functions: C = ( X matroid M α M r M : α M ≥ M ) . This includes weighted matroid rank functions asa subclass. The cone also contains coverage valua-tions, v ( S ) = | S j ∈ S A j | for a set system ( A j ) mj =1 .We also obtain a e (1 − e ) -approximation for thisclass. • Bipartite matching with a matroid constraint: Let G = ( L, R, E ) be a bipartite graph with non-negative weight w j,k on the edge ( j, k ) ∈ E , vertexset on the left-hand side L = [ n ] and a matroid M defined on the vertex set on the right side R .For S ⊆ L , let v ( S ) be the maximum weight of amatching such that vertices matched on the left area subset of S , and the vertices matched on the rightare an independent set in M .It is known that these valuations satisfy the prop-erty of gross substitutes (a subclass of submodularfunctions for which the welfare maximization prob-lem can be solved optimally). In fact there is ap-parently no known example of a gross-substitutesfunction which is not in this form. We still obtain a factor of e (1 − e ) for this class,as well as the cone generated by such functions. • More generally, we can handle classes of (non-submodular) valuations that support a certain formof contention resolution . As an example, we showthat it is possible to obtain a Θ(1 /k )-approximationfor valuations defined by a k -dimensional matchingproblem with matroid constraints in Appendix A.Similarly we can also handle the cone generated bysuch functions. Based on Kazuo Murota’s problem set [18] and personalcommunication with Renato Paes Leme’s tutorial [17]
Our approach isbased on a convex programming relaxation from [2],which relies on properties of stable polynomials and wasinspired by Gurvits’ proof of the van der Waerden con-jecture for the permanent of doubly stochastic matrices.We extend this relaxation to the classes of submodularfunctions that we consider, and prove that its integral-ity gap is bounded by e (1 − e ) . This factor arises fromtwo sources: The first one is a generalization of Gurvits’lemma [13], developed in [2], which yields a factor of e .The second ingredient is the use of a contention reso-lution scheme for matroids, which we use twice to dealwith the presence of matroid rank functions as well asthe assignment constraint of the problem. Combiningthese ingredients via the FKG inequality yields a factorof e (1 − e ) .Technically, we prove that a certain rounding tech-nique achieves a c n approximation factor in terms ofthe expected product of valuations E [ Q ni =1 v i ( x i )]. Thisproves that there exists a solution whose value in termsof the geometric average ( Q ni =1 v i ( x i )) /n is at least c · OP T . However, the rounding method is randomizedand a solution of value c · OP T might appear with anexponentially small probability so the expect runningtime would be exponential.A similar issue already appears in [2], where theauthors resolve this by a method of conditional expec-tations. The estimator of conditional expectations turnsout to be a matching-counting problem which is knownto admit an FPTAS by Markov Chain Monte Carlomethods. In our case, an analogous approach leads tocounting problems which we currently don’t know howto resolve. In particular, in the case of coverage valua-tions, we would need to resolve the following countingproblem.
Open problem: Counting of constrainedmappings.
Given sets
A, B , where B is partitionedinto blocks B , . . . , B n , and weights w ij ≥ i ∈ A, j ∈ B , for each choice of e ∈ B , e ∈ B , . . . , e n ∈ B n let S ( e , . . . , e n ) be the set of all map-pings σ : A → B such that e , . . . , e n ∈ σ [ A ]. Estimatethe quantity X e ∈ B ,...,e n ∈ B n X σ ∈S ( e ,...,e n ) Y i ∈ A w i,σ ( i ) . Finally, we want to remark that the situation issomewhat reminiscent of the Santa Claus problem, forwhich the first constant-factor integrality gap boundwas proved by Uriel Fiege in [8]. This proof relied onthe use of the Lov´asz Local Lemma, which was latermade algorithmically efficient in [14]. While our analysisdoes not use the Lov´asz Local Lemma, there are certainsimilarities here. In some special cases, the rounding
Copyright © ethod essentially attempts to build a perfect matchingby selecting a random match for each vertex. Theprobability of success is obviously exponentially small. We present first our algorithm for (weighted) matroidrank functions, which is technically the simplest casebut it already illustrates the basic ideas of our approach.Further generalizations require some technicalities butmostly follow along the same lines.Let the valuation of player i be a weighted matroidrank function, v i ( x i ) = max { X j ∈ I w ij : I ∈ I i , I ≤ x i } where M i = ([ m ] , I i ) is a matroid and w ij ≥ i ∈ [ n ] , j ∈ [ m ].Let P ( M i ) denote the associated matroid polytope, P ( M i ) = conv( { I : I ∈ I i } ) . It is known that the weighted rank function can beequivalently written as v i ( S ) = max { X j ∈ S w ij x ij : x i ∈ P ( M i ) } . With this in mind, we suggest the following relaxation(an extension of the convex/concave program from [2]).In the rest of the paper, we will also use similarnotations as [2]), denoting Q i ∈ S y i by y S and { S ⊆ [ m ] (cid:12)(cid:12) | S | = n } by (cid:0) [ m ] n (cid:1) . ( P ) max x ∈ R n × m inf y ∈ R m + : y S ≥ , ∀ S ∈ ( [ m ] n ) n Y i =1 m X j =1 w ij x ij y j ,s.t. x i ∈ P ( M i ) ∀ i ∈ [ n ] n X i =1 x ij ≤ ∀ j ∈ [ m ]This is indeed a relaxation, as the next lemma shows. Lemma 2.1.
The optimal solution of the ( P ) relax-ation is at least the optimal solution of the Nash welfaremaximization problem. Proof. Suppose that x ∈ { , } n × m is the optimalallocation. Since the valuation for player i is a weightedmatroid rank function for M i , we can assume withoutloss of generality that the set allocated to player i isindependent in M i (if necessary, remove some items toretain only the independent set that defines the value of v i ( x i )). Then we have x i ∈ P ( M i ) and P ni =1 x ij ≤ y ∈ R m + such that y S ≥ S ∈ (cid:0) [ m ] n (cid:1) , we have n Y i =1 m X j =1 w ij x ij y j = m X j ,...,j n =1 n Y i =1 ( w ij i x ij i y j i )= X distinct j ,...,j n n Y i =1 y j i n Y i =1 ( w ij i x ij i ) ≥ X distinct j ,...,j n n Y i =1 ( w ij i x ij i )= n Y i =1 n X j =1 w ij x ij where the inequality holds because of the constraint y S ≥
1, and the equality between summations over all n − tuples and distinct n -tuples holds because we cannothave x ij = x i ′ j ′ = 1 for i = i ′ and j = j ′ . Therefore,the optimum of the relaxation is at least the integeroptimum.Next, we use the following lemma from [2], whichexplains why the relaxation can be solved efficiently. Lemma 2.2.
The logarithm of the objective function, n X i =1 log m X j =1 w ij x ij y j is concave in x ij and convex in log y j . The added ingredient here is the matroid polytopeconstraint; however as is well known, matroid polytopesadmit efficient separation oracles, and hence our relax-ation can still be solved (within an arbitrarily small er-ror) by standard techniques just like in [2].
Themain remaining question is how to round a fractionalsolution of ( P ). In [2], items are allocated to playersindependently with probabilities x ij , which is a naturalchoice. However, here it is not clear if this works due tothe nonlinearity of the valuation functions. We need oneadditional ingredient, which is contention resolution [5]. Copyright © efinition 2.1. A ( b, c ) -balanced contention resolu-tion for an independence system M = ([ m ] , I ) (e.g. amatroid) is a procedure that for every x ∈ b · P ( M ) and A ⊆ [ m ] returns a (random) set π x ( A ) ⊆ A such that • π x ( A ) ∈ I with probability , and • if R ( x ) is a random set where each element j appears independently with probability x j , then Pr[ j ∈ π x ( R ( x )) | j ∈ R ( x )] ≥ c. The scheme is said to be monotone if
Pr[ i ∈ π x ( A )] ≥ Pr[ i ∈ π x ( A )] whenever i ∈ A ⊆ A . Contention resolution schemes are known now forvarious types of constraints. We mention the followingresult, most relevant to our work here [5].
Theorem 2.1.
For any matroid and any b ∈ (0 , ,there is a monotone ( b, − e − b b ) -balanced contention res-olution scheme. We are going to use contention resolution to makesure that every player receives a set independent intheir respective matroid. We need to proceed carefully,since we need to combine contention resolution withthe analysis of the product of valuation functions, soany dependencies between players (as in the roundingprocedure of [2]) are potentially dangerous. We designthe rounding procedure as follows.
Rounding Procedure 1 : Rounding for ma-troid rank functions.
Given a fractional solution x ∈ R n × m , • For each ( i, j ) ∈ [ n ] × [ m ] independently, set X ij = 1with probability x ij and X ij = 0 otherwise. • For each j ∈ [ m ] independently, apply contentionresolution in the uniform rank-1 matroid on [ n ] tothe set P j = { i : X ij = 1 } , to obtain a singleton { p j } (the player tentatively receiving item j ). • For each i ∈ [ n ] independently, apply contentionresolution in matroid M i to the set S i = { j : X ij =1 } , to obtain an independent set I i ∈ I i (the settentatively allocated to player i ). • Allocate item j to player p j , if j ∈ I i .We remark that the 3rd bullet point and the con-dition j ∈ I i are not crucial for the algorithm to work(and in fact the algorithm might potentially lose somevalue by applying it). However, it ensures that eachplayer receives an independent set, which will be usefulin the analysis. We now proceed to analyze the algorithm. Let usset some notation. Let Y ij denote the indicator variablefor the event that p j = i (i.e., the ( i, j ) pair survives thecontention resolution in the rank-1 matroid). Let Z ij denote the indicator variable for the event that j ∈ S i (i.e., the ( i, j ) pair survives the contention resolution inmatroid M i ). Observe that Y ij Z ij is the indicator ofthe event that player i actually receives item j . Lemma 2.3.
Let j , . . . , j n ∈ [ m ] be distinct items.Then E " n Y i =1 ( Y ij i Z ij i ) ≥ (1 − /e ) n n Y i =1 x ij i . Proof.
Let X denote the event that X ij i = 1 forevery i ∈ [ n ]. This is a necessary condition for Q ni =1 ( Y ij i Z ij i ) = 1 to happen, and by independence,Pr[ X ] = Q ni =1 x ij i . Hence our goal is to prove that E [ Q ni =1 ( Y ij i Z ij i ) | X ] ≥ (1 − /e ) n .Note that for each i ′ , X i ′ j i ′ is the only variableamong ( X ij i ) ni =1 that participates in the contentionresolution scheme in M i ′ , and similarly it is the onlyvariable that participates in the contention resolutionscheme on { ( i, j i ′ ) : 1 ≤ i ≤ n } . Therefore, forthe purposes of these contention resolution schemes,conditioning in X is equivalent to conditioning on X i ′ j i ′ = 1. By the properties of contention resolutionin matroids, we have Pr[ Y ij i = 1 | X ] ≥ − /e and Pr[ Z ij i = 1 | X ] ≥ − /e . However, there aredependencies between these events for different values of i (since conditioning on Y ij , Z ij gives information aboutother elements participating in the contention resolutionschemes), so the lemma doesn’t follow immediately.We need the additional property that our con-tention resolution is monotone , that is elements are lesslikely to survive when selected from a larger set. Moreformally, for P ⊆ [ n ] such that i ∈ P , and S ⊆ [ m ] suchthat j ∈ S , let π ij ( P ) = Pr[ Y ij = 1 | X & P j = P ] ,σ ij ( S ) = Pr[ Z ij = 1 | X & S i = S ] . From Theorem 2.1, π ij ( P ) is a non-increasing functionof P (among sets P such that i ∈ P ), and σ ij ( S ) is anon-increasing function of S (among sets S such that j ∈ S ). Also, note that once the variables X ij are fixed,the sets P j , S i are determined, and each contentionresolution procedure proceeds independently. Hence, wehave E " n Y i =1 Y ij i Z ij i | X = E " n Y i =1 π ij i ( P j i ) σ ij i ( S i ) | X . Copyright © ow we invoke the FKG inequality: Fornon-decreasing functions π ( ω ) , σ ( ω ) on a productspace Ω with a product measure, E Ω [ π ( ω ) σ ( ω )] ≥ E Ω [ π ( ω )] E Ω [ σ ( ω )]. Here, the product space is the spaceof random variables X ij conditioned on X (which stillkeeps the remaining random varables { X ij , j = j i } in-dependent). Hence, by applying the FKG inequalityrepeatedly, E " n Y i =1 π ij i ( P j i ) σ ij i ( S i ) | X ≥ n Y i =1 ( E [ π ij i ( P j i ) | X ] · E [ σ ij i ( S i ) | X ]) . Finally, from Theorem 2.1, we get E P j [ π ij ( P j ) | X ] = Pr[ Y ij = 1 | X ] ≥ − /e, E S i [ σ ij ( S i ) | X ] = Pr[ Z ij = 1 | X ] ≥ − /e. This completes the proof that E [ Q ni =1 ( Y ij i Z ij i ) | X ] ≥ (1 − /e ) n .Now we complete the analysis of the roundingprocedure. We need the following key lemma from [2](a generalization of a lemma by Gurvits). Lemma 2.4.
Let p ( y , . . . , y m ) be a homogeneousdegree- n real stable polynomial with nonnegative coef-ficients. For S ⊆ [ m ] , let c S denote the coefficient ofthe monomial y S . Then X S ∈ ( [ m ] n ) c S ≥ e − n inf y> ∀ S ∈ ( [ m ] n ) ,y S ≥ p ( y , . . . , y m ) . We are going to apply this lemma to the polynomial p ( y , . . . , y m ) = Q ni =1 (cid:16)P mj =1 w ij x ij y j (cid:17) which is real-stable (just like in [2]). We prove the following. Theorem 2.2.
The outcome of Rounding Procedure 1has expected value at least e (1 − e ) OP T .Proof.
Denote by J i the set allocated to player i , i.e. J i = { j ∈ I i : p j = i } . Since J i ⊆ I i ∈ I i , J i is also in I i ; hence the value obtained by player i is v i ( J i ) = P j ∈ J i w ij . It remains to analyze the expectedvalue of the product Q ni =1 v i ( J i ). E " n Y i =1 v i ( J i ) = E n Y i =1 X j ∈ J i w ij = E n Y i =1 m X j =1 w ij Y ij Z ij where Y ij , Z ij are the indicator variables of j ∈ I i and p j = i as above. We continue, E n Y i =1 m X j =1 w ij Y ij Z ij = m X j ,...,j n =1 E " n Y i =1 w ij i Y ij i Z ij i ≥ X distinct j ,...,j n E " n Y i =1 w ij i Y ij i Z ij i . (The last inequality is in fact an equality, because thesame item cannot be allocated to multiple players, butwe don’t need that here.) We appeal to Lemma 2.3: Forevery choice of distinct j , . . . , j n , E [ Q ni =1 Y ij i Z ij i ] ≥ (1 − /e ) n Q ni =1 x ij i . Therefore, X distinct j ,...,j n E " n Y i =1 w ij i Y ij i Z ij i ≥ (1 − /e ) n X distinct j ,...,j n n Y i =1 w ij i x ij i . The last expression is exactly (1 − /e ) n times thesummation of coefficients for monomials y S , S ∈ (cid:0) [ m ] n (cid:1) inthe polynomial p ( y , . . . , y m ) = Q ni =1 (cid:16)P mj =1 w ij x ij y j (cid:17) .By Lemma 2.4, X distinct j ,...,j n n Y i =1 w ij i x ij i ≥ e − n inf y> ∀ S ∈ ( [ m ] n ) ,y S ≥ n Y i =1 m X j =1 w ij x ij y j = e − n OP T for an optimal fractional solution x . We conclude that E " n Y i =1 v i ( J i ) ≥ e − n (1 − /e ) n OP T.
In this section we extend our result to the cone gen-erated by matroid rank functions, or equivalently thesums of weighted matroid rank functions. Consider therounding method that skips the third bullet point inRounding 1.
Rounding Procedure 2 :
Given a fractionalsolution x ∈ R n × m , • For each ( i, j ) ∈ [ n ] × [ m ] independently, set X ij = 1with probability x ij and X ij = 0 otherwise. Copyright © For each j ∈ [ m ] independently, apply contentionresolution in the uniform rank-1 matroid on [ n ] tothe set P j = { i : X ij = 1 } , to obtain a singleton { p j } (the player tentatively receiving item j ). • Allocate item j to player p j .It’s easy to see that the expected value of theobjective function after this new rounding is at leastthe expected value after the original rounding. Lemma 2.5.
For any weighted matroid rank functions,the outcome of Rounding Procedure 2 is at least theoutcome of Rounding Procedure 1.Proof.
Obviously, the set allocated to each agent i in Rounding Procedure 2 contains the set allocatedin Rounding Procedure 1. By monotonicity of thevaluations, the value obtained by Rounding Procedure2 dominates the value obtained by Rounding Procedure1. An interesting point about this new rounding isthat it only depends on the fractional solution, inother words, oblivious to the matroids. So we can usethis rounding more generally for sums of matroid rankfunctions. Let the valuation of agent i be the summationof several weighted matroid rank functions, v i ( x i ) = s i X k =1 α k v ik ( x i )where v ik ( x i ) = max { X j ∈ I ω ijk : I ∈ L ik , I ≤ x i } . where M ik = ([ m ] , L ik ) is a matroid and ω ijk ≥ i ∈ [ n ], j ∈ [ m ], k ∈ [ s i ]. Similarly let P ( M ik ) bethe associated matroid polytope, we have the followingrelaxation. Note that z i ∗ k is the vector with m elements( z ijk ) mj =1 .max x , z inf y ∈ R m + : y S ≥ , ∀ S ∈ ( [ m ] n ) n Y i =1 s i X k =1 m X j =1 w ijk z ijk y j ,s.t. z i ∗ k ∈ P ( M ik ) ∀ i ∈ [ n ] , k ∈ [ s i ] n X i =1 x ij ≤ ∀ j ∈ [ m ] z ijk ≤ x ij ∀ i ∈ [ n ] , ∀ j ∈ [ m ] , ∀ k ∈ [ s i ]It is easy to prove this is a relaxation of the Nashwelfare maximization problem. Lemma 2.6.
The optimal solution of the above programis at least the optimal solution of the Nash SocialWelfare maximization problem.Proof.
Suppose that x ∈ { , } n × m is the optimalallocation. Let z i ∗ k ∈ { , } m be the indicator of amaximal independent set in matroid M ik such that z i ∗ k ≤ x i . Then we have z i ∗ k ∈ P ( M ik ), P ni =1 x ij ≤ z ijk ≤ x ij . Also, for every y ∈ R m + such that y S ≥ S ∈ (cid:0) [ m ] n (cid:1) , we have n Y i =1 s i X k =1 m X j =1 w ij z ijk y j = X k ,...,k n ≤ k i ≤ s i m X j ,...,j n =1 n Y i =1 ( w ij i k i z ij i k i y j i )= X k ,...,k n ≤ k i ≤ s i X distinct j ,...,j n y { j ,...,j n } n Y i =1 ( w ij i k i z ij i k i ) ≥ X k ,...,k n ≤ k i ≤ s i X distinct j ,...,j n n Y i =1 ( w ij i k i z ij i k i )= n Y i =1 s i X k =1 m X j =1 w ijk i z ijk where the inequality holds because of the constraint y S ≥
1, and the equality between summations over all n − tuples and distinct n -tuples holds because we cannothave z ijk = z i ′ j ′ k ′ = 1 for i = i ′ and j = j ′ (each itemis assigned to only one player). Therefore, the optimumof the above program is at least the integer optimum.Moreover, similarly, by Lemma 2.2 and standardtechniques using separation oracles for matroid poly-topes, we can solve this program efficiently (to arbitraryprecision). Finally, we use the Rounding Procedure 2 toround a fractional solution of the program. Theorem 2.3.
The outcome of the Rounding 2 hasexpected value at least e (1 − e ) OP T .Proof.
Denote by J i the set allocated to player i . Fix k , . . . , k n such that 1 ≤ k i ≤ s i , consider the expectedvalue of E [ Q ni =1 v ik i ( J i )] after the rounding. Since x ij ≥ z ijk , we know that the outcome of assigningeach item with probability x ij is at least the outcome ofassigning each item with probability z ijk i . Moreover, byLemma 2.5, we know that the outcome can be furtherlower-bounded by the outcome of Rounding Procedure Copyright © , which is at least(1 − /e ) n X j ,...,j n distinct n Y i =1 w ij i k i z ij i k i . by Theorem 2.2. Putting all the possible choices of k , . . . , k n together, we have E " n Y i =1 s i X k i =1 v ik i ( J i ) = X k ,...,k n ≤ k i ≤ s i E " n Y i =1 v ik i ( J i ) ≥ X k ,...,k n ≤ k i ≤ s i (1 − /e ) n X j ,...,j n distinct n Y i =1 w ij i k i z ij i k i =(1 − /e ) n X j ,...,j n distinct n Y i =1 s i X k =1 w ij i k z ij i k ≥ e − n (1 − /e ) n inf y ∈ R m + : y S ≥ , ∀ S ∈ ( [ m ] n ) n Y i =1 s i X k =1 m X j =1 w ijk z ijk y j where the last inequality is due to Lemma 2.4. Finally,due to Lemma 2.6, the last quantity is at least e (1 − e ) OP T . In this section we discuss the case of valuation func-tions defined by a bipartite matching problem with amatroid constraint. More specifically, consider n bipar-tite graphs G i = ( L i , R i , E i ), 1 ≤ i ≤ n , L i = [ m ], witha non-negative weight w i,j,k on each edge ( j, k ) ∈ E i ,and a matroid M i = ( R i , I i ) defined on on the right-hand side R i of each graph. Denote by v i the valuationfunction for player i . For any S ⊆ L i = [ m ], we define v i ( S ) to be the maximum weight of a matching M i suchthat vertices matched on the left are a subset of S , andthe vertices matched on the right are an independentset in M i .Denote by P ( M i ) the associated matroid polytope.Let z ∈ R E × . . . × R E n be a vector indexed by edges ofthe bipartite graphs G i where z i,j,k represents whetheredge ( j, k ) ∈ E i is in the matching M i or not. Considerthe following mathematical program, max z ∈ R E × ... × R En inf y > y S ≥ ∀ S ∈ ( [ m ] n ) n Y i =1 X ( j,k ) ∈ E i w i,j,k z i,j,k y j s.t. X i,k :( j,k ) ∈ E i z i,j,k ≤ ∀ j ∈ [ m ] u i,k = X j :( j,k ) ∈ E i z i,j,k ≤ ∀ ≤ i ≤ n, k ∈ R i u i ∈ P ( M i ) ∀ ≤ i ≤ n ≤ z i,j,k ≤ ∀ ≤ i ≤ n, ( j, k ) ∈ E i First we show that the program above is indeed arelaxation of Nash Social Welfare maximization.
Lemma 3.1.
The optimal solution of the above programis at least the optimal solution of the Nash SocialWelfare maximization problem.Proof.
Suppose that x ∈ { , } n × m is the optimalallocation. Assume one of the maximal matchingsin graph G i subject to the matroid constraint M i ,corresponding to this assignment, is z i ∈ { , } E i . Thenit is easy to verify that z i satisfies all the constraints inthe above program. Also, for every y ∈ R m + such that y S ≥ S ∈ (cid:0) [ m ] n (cid:1) , we have n Y i =1 X ( j,k ) ∈ E i w i,j,k x i,j,k y j = X ( j i ,k i ) ∈ E i ≤ i ≤ n n Y i =1 ( w i,j i ,k i x i,j i ,k i y j i )= X distinct j ,...,j n ( j i ,k i ) ∈ E i y { j ,...,j n } n Y i =1 ( w i,j i ,k i x i,j i ,k i ) ≥ X distinct j ,...,j n ( j i ,k i ) ∈ E i n Y i =1 ( w i,j i ,k i x i,j i ,k i )= n Y i =1 X ( j,k ) ∈ E i w ijk x ijk where the inequality holds because of the constraint y S ≥
1, and the equality between summations over all n − tuples and distinct n -tuples holds because we cannothave z i,j,k = z i ′ ,j ′ ,k ′ = 1 for i = i ′ and j = j ′ (we assignan item to only one player). Therefore, the optimumof the above program is at least the integer optimum. Copyright © imilarly, by Lemma 2.2 and standard techniqueslike [2] using separation oracles for matroid polytopeswe can solve this program efficiently.Next we are going to use contention resolution to de-sign a rounding procedure. Notice that the constraintsfor z correspond to the polytope of the intersection oftwo matroids P M ( L ) ∩M ( R ) : one partition matroid M ( L ) encoding that at most 1 edge can be chosen among allthe edges incident to the same item j (across all thegraphs G i ); and another (general) matroid M ( R ) suchthat among the edges in each G i , at most one edge in-cident to each vertex on the right can be chosen, andthe matched vertices form an independent set in M i .( M ( R ) can be constructed by taking a disjoint union ofthe matroids M i , and adding parallel copies of elementsrepresenting the different edges incident to a given ver-tex on the right.)We can further define the allocation constraint onitem j as M ( L ) j and the matroid constraint on the right-hand side of graph G i as M ( R ) i . Our rounding algorithmis as follows. Rounding Procedure 3 : Rounding for bi-partite matching with matroid constraints valu-ations.
Given a fractional solution z , • For each edge ( j, k ) ∈ E i , 1 ≤ i ≤ n independently,set X ijk = 1 with probability z i,j,k and X i,j,k = 0otherwise. • For each item j ∈ [ m ] independently, run a (1 , − e − )-balanced contention resolution algorithm onthe set P j = { ( i ′ , j ′ , k ′ ) : j ′ = j, X i ′ ,j ′ ,k ′ = 1 } and with the matroid M ( L ) j . Denote by Y i,j,k theindicator random variable of the event that edge( i, j, k ) is selected in this contention resolution. • For each agent i ∈ [ n ] independently, run a (1 , − e − )-balanced contention resolution algorithm onthe set Q i = { ( i ′ , j ′ , k ′ ) : i ′ = i, X i ′ ,j ′ ,k ′ = 1 } and with the matroid M ( R ) i . Denote by Z i,j,k theindicator random variables of the event that edge( i, j, k ) is selected in this contention resolution. • Allocate item j to agent i , if Y i,j,k Z i,j,k = 1 forsome k ∈ R i .We can prove that this rounding algorithm is a goodapproximation using a similar strategy as Theorem 2.2. Theorem 3.1.
The outcome of the rounding above hasexpected value at least e (1 − e ) OP T .Proof.
First we will prove a similar inequality asLemma 2.3: consider n distinct items j , . . . , j n where j i is allocated to player i . That means there must besome edge ( j i , k i ) chosen in the rounding. We will showthe following inequality: E " n Y i =1 Y i,j i ,k i Z i,j i ,k i ≥ (1 − e ) n n Y i =1 z i,j i ,k i As we have discussed in the rounding procedure,for any edge ( j, k ) ∈ E i , Y i,j,k is a function of variablesin P j , and Z i,j,k is a function of variables in Q i .Furthermore, by the monotonicity of the contentionresolution scheme, E [ Y i,j,k | X i ′ ,j,k ′ : ( i ′ , j, k ′ ) ∈ P j ]and E [ Z i,j,k | X i,j ′ ,k ′ : ( i, j ′ , k ′ ) ∈ Q i ]are both functions that are non-increasing in the vari-ables X i ′ ,j ′ ,k ′ , except X i,j,k itself which must be 1, oth-erwise Y i,j,k = Z i,j,k = 0.Let X denote the event that ∀ i ∈ [ n ], X i,j i ,k i = 1.As in the previous section, we have a product proba-bility space conditioned on X , and non-increasing func-tions on this product space. By repeated applicationsof the FKG inequality, we have E " n Y i =1 Y i,j i ,k i Z i,j i ,k i |X ≥ n Y i =1 E [ Y i,j i ,k i |X ] E [ Z i,j i ,k i |X ] . By the properties of contention resolution schemewe use, for any 1 ≤ i ≤ n , E [ Y i,j i ,k i |X ] = E [ Y i,j i ,k i | X i,j i ,k i ] ≥ − /e, E [ Z i,j i ,k i |X ] = E [ Z i,j i ,k i | X i,j i ,k i ] ≥ − /e. Putting this together we have E " n Y i =1 Y i,j i ,k i Z i,j i ,k i = E " n Y i =1 Y i,j i ,k i Z i,j i ,k i |X Pr[ X ] ≥ (1 − e ) n Pr[ X ] = (1 − e ) n n Y i =1 z i,j i ,k i . Now we can bound the expect product of valuation
Copyright © ver all agents, after the rounding, which is X distinct j ,...,j n ( j i ,k i ) ∈ E i E " n Y i =1 Y i,j i ,k i Z i,j i ,k i ≥ (1 − e ) n X distinct j ,...,j n ( j i ,k i ) ∈ E i n Y i =1 z i,j,k . By Lemma 2.4 (Generalized Gurvits’ Lemma), weknow that is at least(1 − e ) n e − n inf y > y S ≥ , ∀ S ∈ ( [ m ] n ) n Y i =1 X ( j,k ) ∈ E i w i.j,k z i,j,k y j which is a (1 − /e ) n e − n approximation for the objec-tive function.Moreover, similar to the case of weighted matroidrank functions, we can prove that skipping the thirdbullet point in the rounding will not decrease theoutcome and the procedure will not depend on thevaluation functions any more. So similarly, we canstill use the new rounding method for sums of bipartitematching valuations with a matroid constraint. Theproof is quite similar to the matroid case and we omitthe proof here. References [1] Nima Anari, Tung Mai, Shayan Oveis Gharan, andVijay V. Vazirani. Nash social welfare for indivisibleitems under separable, piecewise-linear concave utili-ties. In
Proceedings of the 29th Annual ACM-SIAMSymposium on Discrete Algorithms , pages 2274–2290.ACM, January 2018.[2] Nima Anari, Shayan Oveis Gharan, Amin Saberi, andMohit Singh. Nash social welfare, matrix perma-nent, and stable polynomials. In . Schloss Dagstuhl-Leibniz-Zentrum fuer Infor-matik, 2017.[3] Siddharth Barman, Sanath Kumar Krishnamurthy,and Rohit Vaish. Finding fair and efficient allocations.In
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Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Al-gorithms , pages 2326–2340. SIAM, 2018.[12] Jugal Garg, Pooja Kulkarni, and Rucha Kulkarni.Approximating nash social welfare under submodularvaluations through (un) matchings. In
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Copyright © High Dimensional matching with matroidconstraints
In this section we consider the most general class ofvaluation functions we can handle – high dimensionalmatching with matroid constraints.Let us represent the valuation function of eachplayer by a weighted k-partite hypergraph and k − i , consider a hypergraph G i =( V i , E i ) where V i = S i ∪ T i, ∪ . . . ∪ T i,k − , S i = [ m ] is theset of items, and T i, , . . . , T i,k − are disjoint sets; eachhyperedge in E i contains exactly 1 vertex from eachof S i , T i, , . . . , T i,k − . We have a non-negative weight w ( i ) e on the hyperedge e ∈ E i , and a matroid constraint M i,ℓ = ( T i,ℓ , I i,ℓ ) defined on each T i,ℓ for 1 ≤ ℓ ≤ k .Denote by v i the valuation function for agent i . For any S ⊆ S i = [ m ], we define v i ( S ) to be the maximumweight of a matching of hyperedges in G i such thatvertices matched in S i are a subset of S , and the verticesmatched in T i,ℓ are an independent set in M i,ℓ for every1 ≤ ℓ ≤ k − P ( M i,ℓ ) the matroid polytope for ma-troid M i,ℓ . Let z ∈ R E × ... × R E n be an allocationvector where z ( i ) e = 1 indicates that edge e ∈ E i is inthe matching representing the value of v i ( S ). Considerthe following mathematical program:max z ∈ R E × ... × R En inf y > y S ≥ ∀ S ∈ ( [ m ] n ) n Y i =1 X e ∈ E i e i incident on j in S i w ( i ) e z ( i ) e y j s.t. X ≤ i ≤ n,e ∈ E i z ( i ) e ≤ , ∀ j ∈ [ m ] u i,ℓv = X e ∈ E i : v ∈ e z ( i ) e ≤ , ∀ ≤ i ≤ n, ≤ ℓ ≤ k, v ∈ T i,ℓ u i,ℓ ∈ P ( M i,ℓ ) , ∀ ≤ i ≤ n, ≤ ℓ ≤ k ≤ z ( i ) e ≤ , ∀ ≤ i ≤ n, e ∈ E i First we prove that the program above is indeed arelaxation of Nash social welfare maximization.
Lemma A.1.
The optimal solution of the above programis at least the optimal solution of the Nash SocialWelfare problem.Proof.
Suppose that x ∈ { , } n × m is the optimalallocation. Let z i ∈ { , } | E i | denote a maximum-weight matching in G i attaining the value of agent i andsatisfying the respective matroid constraints. It is easyto verify that z i satisfies all the constraints in the aboveprogram. Also, for every y ∈ R m + such that y S ≥ S ∈ (cid:0) [ m ] n (cid:1) , we have n Y i =1 X j ∈ S i X e ∈ E i : e ∩ S i = { j } w ( i ) e z ( i ) e y j = X j ∈ S ,...,j n ∈ S n X e i ∈ E i : e i ∩ S i = { j i } n Y i =1 (cid:16) w ( i ) e i z ( i ) e i y j i (cid:17) = X distinct j ,...,j n e i ∈ E i : e i ∩ S i = { j i } y { j ,...,j n } n Y i =1 (cid:16) w ( i ) e i z ( i ) e i (cid:17) ≥ X distinct j ,...,j n e i ∈ E i : e i ∩ S i = { j i } n Y i =1 (cid:16) w ( i ) e i z ( i ) e i (cid:17) = n Y i =1 X e ∈ E i w ( i ) e z ( i ) e ! where the inequality holds because of the constraint y S ≥
1, and the equality between summations over all n − tuples and distinct n -tuples holds because we cannothave z ( i ) e i = z ( i ′ ) e i ′ = 1 for i = i ′ , e i and e i ′ incident onthe same item j . Therefore, the optimum of the aboveprogram is at least the integer optimum.Similarly, by Lemma 2.2 and standard convex op-timization techniques, using separation oracles for ma-troid polytopes, we can solve this program efficiently.Next we are going to use contention resolutionscheme to design a rounding procedure. Notice that theconstraints for variable z actually corresponds to theintersection of k matroid polytopes ∩ k − i =0 P ( M ( i ) ): onepartition matroid M (0) for the assignment constraint(vertex sets V , , . . . , V n, ) saying that each item isallocated to at most one agent and that an agent canallocate at most one edge to this item, a general matroidconstraint M ( ℓ ) on V ,ℓ , . . . , V n,ℓ for any 1 ≤ ℓ ≤ k − i , the matchedsubset of each T i,ℓ is independent in M i,ℓ .Notice that each of those matroids is a union of n matroids, one for each agent. The partition ma-troid M (0) can be split it up as M (0) = S mj =1 M (0) j where M (0) j expresses the allocation constraint for item j . For the other matroids M ( ℓ ) , we can write M ( ℓ ) = ∪ ni =1 M ( ℓ ) i where M ( ℓ ) i expresses the ℓ -th matroid con-straint for agent i . Our rounding algorithm works asfollows. Rounding Procedure 4 : Rounding fork-partite matching with matroid constraints.
Given a fractional solution z , • For every agent 1 ≤ i ≤ n and each edge e ∈ E i ,independently set X ( i ) e = 1 with probability z ( i ) e and X ( i ) e = 0 otherwise. Copyright © For each item j ∈ [ m ], independently run a (1 , − /e ) contention resolution algorithm on the set P j = { ( e, i ) : e ∈ E i , ≤ i ≤ n, X ( i ) e = 1 , j ∈ e } with the matroid constraint M (0) j . Denote by Y ( i ) e the indicator random variable for the event thatedge e ∈ E i is selected in this procedure. • For each agent i ∈ [ n ] and 1 ≤ ℓ ≤ k −
1, independently run a ( b, b (1 − e − b )) contentionresolution algorithm for b = k − on the set Q i = { e ∈ E i : X ( i ) e = 1 } with the matroid constraint M ( i ) ℓ . Denote by Z ( i ) e,ℓ = 1 the indicator randomvariable for the event that e ∈ E i is selected in thisprocedure. • We allocate item j to agent i , if there is edge e ∈ E i incident to this item j such that Y ( i ) e Z ( i ) e = 1 where Z ( i ) e = Q kℓ =1 Z ( i ) e,ℓ .We can prove that this rounding algorithm is a goodapproximation using a similar strategy as Theorem 2.2. Theorem A.1.
The outcome of Rounding Procedure 4has expected value at least ( e k ) − n OP T .Proof.
First we will prove a similar inequality asLemma 2.3: consider n distinct items j , . . . , j n where j i is allocated to player i . That means there must besome edge e i such that it is chosen in the rounding and e i is incident on j i in S i . We will show the followinginequality: E " n Y i =1 Y ( i ) e i Z ( i ) e i ≥ ( ek ) − n n Y i =1 z ( i ) e i As we have discussed in the rounding procedure, forany edge e ∈ E i incident to item j , Y ( i ) e is a function ofvariables in P j , and Z i,j,k is a function of variables in Q i . Furthermore, by the monotonicity of the contentionresolution scheme, E [ Y ( i ) e | X ( i ′ ) e ′ : ( e ′ , i ′ ) ∈ P j ]and E [ Z ( i ) e | X ( i ′ ) e ′ : e ′ ∈ Q i ]are both functions that are decreasing in the variables X ( i ′ ) e ′ , except X ( i ) e itself which must be 1 otherwise Y ( i ) e = Z ( i ) e = 0.Let X denote the event that ∀ i ∈ [ n ], X ( i ) e i = 1. Bythe FKG inequality applied repeatedly, we get E " n Y i =1 Y ( i ) e i Z ( i ) e i |X ≥ n Y i =1 E h Y ( i ) e i |X i E h Z ( i ) e i |X i . By the properties of the contention resolutionschemes we use, for any 1 ≤ i ≤ n , E h Y ( i ) e i |X i = E h Y ( i ) e i | X ( e ) e i ≥ − /e and E h Z ( i ) e i |X i ≥ k − − exp {− k − } / ( k − ! k − ≥ k − − / (1 + k − )1 / ( k − ! k − = 1 k − (cid:18) k − k (cid:19) k − ≥ ek . Putting this together we have E " n Y i =1 Y ( i ) e i Z ( i ) e i = E " n Y i =1 Y ( i ) e i Z ( i ) e i |X Pr[ X ] ≥ ( ek ) − n Pr[ X ] ≥ ( e k ) − n n Y i =1 z ( i ) e i . Now we can bound the product of valuations overall agents, after rounding, which is X distinct j ,...,j n e i ∈ E i : e i ∩ S i = { j i } E " n Y i =1 Y ( i ) e i Z ( i ) e i ≥ ( ek ) − n X distinct j ,...,j n e i ∈ E i : e i ∩ S i = { j i } n Y i =1 z ( i ) e i . By Lemma 2.4 (Generalized Gurvits’ Lemma), weknow that is at least( ek ) − n e − n inf y > y S ≥ , ∀ S ∈ ( [ m ] n ) n Y i =1 X e ∈ E i e i incident on j in S i w ( i ) e z ( i ) e y j which is a ( e k ) − n approximation for objective functionof the program.Finally, similar to the case of weighted matroid rankfunctions, we can show that skipping the third bulletpoint in the rounding will not decrease the outcome andthe procedure will not depend on the valuation functionsany more. So we can still use this new rounding toextend the result to the cone of k -dimensional matchingvaluations with matroid constraints. The proof is quitesimilar to the matroid case, so we omit it here. Copyright © Open counting problems and theirconnection to Nash Social Welfare
Let us discuss here some of the directions that our workleaves open. One way to resolve the issue of finding agood allocation of items might be to apply the methodof conditional expectations. Given a fractional solutionof our concave-convex program, suppose that we applythe following rounding procedure (just like in [2]).
Rounding Procedure 0 : simple rounding.
Given a fractional solution x ∈ R n × m :For each j ∈ [ m ] independently, allocate item j to atmost one agent, agent i with probability x ij . Lemma B.1.
For arbitrary monotone valuations,Rounding Procedure 0 provides expected Nash SocialWelfare at least as large as Rounding Procedure 2.Proof.
We show that the two procedures can be coupledin a way that every agent in Rounding Procedure 2receives a subset of what they receive in RoundingProcedure 0.Consider Rounding Procedure 2: we first set eachvariable X ij independently to be 1 with probability x ij , and 0 otherwise. Then for each fixed j , we usecontention resolution to select one of the variables X ij equal to 1 (if any) and allocate item j to this agent i . An explicit description of such a procedure can befound in [9], Section 1.2: Given initial probabilities x j , . . . , x nj , agent i receives item j with probability x ′ ij = x ij P i x ij (1 − Q i (1 − x ij )) ≥ (1 − /e ) x ij .Now let us consider the event where no agentreceives item j : This occurs with probability 1 − P i x ′ ij = Q i (1 − x ij ). We modify the scheme byallocating item j in case the contention resolutionscheme yielded no allocation, to agent i with additionalprobability x ij − x ′ ij ; with the remaining probability Q i (1 − x ij ) − P i ( x ij − x ′ ij ) = 1 − P i x ij , nobody receivesthe item. Note that the resulting scheme is exactlyRounding Procedure 0: agent i receives item j withprobability x ij , indepenently for each item (since allthe randomness that we used was independent for eachitem j ). As a result, what agents receive in RoundingProcedure 0 dominates what they receive in RoundingProcedure 2.Let us consider the case of coverage, where eachvaluation can be viewed as a summation of rank-1matroid rank functions (corresponding to the coverageof individual elements). Here, the relaxation looks asfollows. ( P ′′ ) max x ∈ R n × m inf y ∈ R m + : y S ≥ , ∀ S ∈ ( [ m ] n ) n Y i =1 X e ∈ U X j ∈ D ie x ije y j ,s.t. X j ∈ D ie x ije ≤ ∀ i ∈ [ n ] , e ∈ Ux ije ≤ x ij ∀ i ∈ [ n ] , e ∈ U, j ∈ D iem X i =1 x ij ≤ ∀ j ∈ [ m ] x ij , x ije ≥ ∀ i ∈ [ n ] , j ∈ [ m ] , e ∈ U Here, the variable x ije denotes the amount to whichitem j covers element e for agent i ; D ie is the set of itemswhose set for agent i covers element e . We considerRounding Procedure 0, where item j goes to agent i with probability x ij . Here, the expected value of therandom assignment is X σ :[ m ] → [ n ] m Y j =1 x σ ( j ) ,j n Y i =1 |{ e ∈ U : ∃ j ∈ D ie , σ ( j ) = i }| = X σ :[ m ] → [ n ] m Y j =1 x σ ( j ) ,j n Y i =1 X e ∈ U ∃ j ∈ D ie ,σ ( j )= i = X e ,...,e n ∈ U X σ :[ m ] → [ n ] m Y j =1 x σ ( j ) ,j n Y i =1 ∃ j ∈ D iei ,σ ( j )= i = X e ,...,e n ∈ U X σ : ∀ i ∃ j ∈ D iei ,σ ( j )= i m Y j =1 x σ ( j ) ,j . This last summation can be viewed as the followingcounting problem: For each agent i , fix an element e i in her coverage universe. Let S ( e , . . . , e n ) be theset of assignments of items such that each agent i receives some item j ∈ D ie i , i.e. some item coveringthe element e i . We want to count the probability thata random assignment satisfies this property, summedup over all choices of e , . . . , e n . It is conceivable thatthis counting problem can be reduced to counting ofbipartite matchings, but we haven’t succeeded in doingso. Copyright ©©