The General Graph Matching Game: Approximate Core
aa r X i v : . [ c s . G T ] J a n The General Graph Matching Game:Approximate Core
Vijay V. Vazirani *11
University of California, Irvine
Abstract
The classic paper of Shapley and Shubik [SS71] characterized the core of the assignmentgame using ideas from matching theory and LP-duality theory and their highly non-trivialinterplay. Whereas the core of the assignment game is always non-empty, that of the generalgraph matching game can be empty.This paper salvages the situation by giving an imputation in the 2/3-approximate core forthe latter. This bound is best possible, since it is the integrality gap of the natural underlyingLP. Our profit allocation method goes further: the multiplier on the profit of an agent lies inthe interval [ , 1 ] , depending on how severely constrained the agent is.The core is a key solution concept in cooperative game theory. It contains all ways ofdistributing the total worth of a game among agents in such a way that no sub-coalition hasincentive to secede from the grand coalition. Our imputation, in the 2/3-approximate core,implies that a sub-coalition will gain at most a 3/2 factor by seceding, and less in typicalcases. * Supported in part by NSF grant CCF-1815901. Introduction
The matching game forms one of the cornerstones of cooperative game theory. This game canalso be viewed as a matching market in which utilities of the agents are stated in monetary termsand side payments are allowed, i.e., it is a transferable utility (TU) market . A key solution conceptin this theory is that of the core , which captures all possible ways of distributing the total worthof a game among individual agents in such a way that the grand coalition remains intact, i.e.,a sub-coalition will not be able to generate more profits by itself and therefore has no incentiveto secede from the grand coalition. For an extensive coverage of these notions, see the book byMoulin [Mou14].When restricted to bipartite graphs, the matching game is called the assignment game . The classicpaper of Shapley and Shubik [SS71] characterized profit-sharing methods that lie in the core ofsuch games by using ideas from matching theory and LP-duality theory and their highly non-trivial interplay; in particular, the core is always non-empty.On the other hand, for games defined over general graphs, the core is not guaranteed to be non-empty, see Section for an easy proof. The purpose of this paper is to salvage the situation to theextent possible by giving a notion of approximate core for such games. The approximation factorwe achieve is 2/3. This is best possible, since it is the integrality gap of the underlying LP; thisfollows easily from an old result of Balinski [Bal65] characterizing the vertices of the polytopedefined by the constraints of this LP. It turns out that the constraints of dual of this LP must berespected by any profit-sharing mechanism.An interesting feature of our profit-sharing mechanism is that it restricts only the most severelyconstrained agents to a multiplier of 2/3, and the less severely constrained an agent is, the betteris her multiplier, all the way to 1; bipartite graphs belong to the last category. One way of statingthe improved factor is: if the underlying graph has no odd cycles of length less than 2 k +
1, thenour factor is k k + .The following setting, taken from [EK01] and [BKP12], vividly captures the underlying issues.Suppose a tennis club has a set V of players who can play in an upcoming doubles tournament.Let G = ( V , E ) be a graph whose vertices are the players and an edge ( i , j ) represents the factthat players i and j are compatible doubles partners. Let w be an edge-weight function for G ,where w ij represents the expected earnings if i and j do partner in the tournament. Then the totalworth of agents in V is the weight of a maximum weight matching in G . Assume that the clubpicks such a matching M for the tournament. The question is how to distribute the total profitamong the agents — strong players, weak players and unmatched players — so that no subset ofplayers feel they will be better off seceding and forming their own tennis club. Definition 1.
The general graph matching game consists of an undirected graph G = ( V , E ) and anedge-weight function w . The vertices i ∈ V are the agents and an edge ( i , j ) represents the factthat agents i and j are eligible for an activity, for concreteness, let us say trading. If ( i , j ) ∈ E , w ij represents the profit generated if i and j trade. The worth of a coalition S ⊆ V is definedto be the maximum profit that can be generated by trades made within S and is denoted by2 ( S ) . Formally, p ( S ) is defined to be the weight of a maximum weight matching in the graph G restricted to vertices in S only. The characteristic function of the matching game is defined to be p : 2 V → R + .Among the possible coalitions, the most important one is of course V , the grand coalition . Definition 2. An imputation gives a way of dividing the worth of the game, p ( V ) , among theagents. Formally, it is a function v : V → R + such that ∑ i ∈ V v ( i ) = p ( V ) . An imputation v issaid to be in the core of the matching game if for any coalition S ⊆ V , there is no way of dividing p ( S ) among the agents in S in such a way that all agents are at least as well off and at least oneagent is strictly better off than in the imputation v .We next describe the characterization of the core of the assignment game given by Shapley andShubik [SS71]. In this game, agents are of two types, buyers B and sellers R . Let G = ( B , R , E ) be a bipartite graph over agent sets B and R , and having edges E whose weights are given by w .The worth of this game, w ( B ∪ R ) , is the weight of a maximum weight matching in G ; the linearprogram (5) gives the LP-relaxation of the problem of finding such a matching. In this program,variable x ij indicates the extent to which edge ( i , j ) is picked in the solution. Matching theorytells us that this LP always has an integral optimal solution; the latter is a maximum weightmatching in G . max ∑ ( i , j ) ∈ E w ij x ij s.t. ∑ ( i , j ) ∈ E x ij ≤ ∀ i ∈ B , ∑ ( i , j ) ∈ E x ij ≤ ∀ j ∈ R , x ij ≥ ∀ ( i , j ) ∈ E (1)Taking v i and u j to be the dual variables for the first and second constraints of (1), we obtain thedual LP: min ∑ i ∈ B v i + ∑ j ∈ R u j s.t. v i + u j ≥ w ij ∀ ( i , j ) ∈ E , v i ≥ ∀ i ∈ B , u j ≥ ∀ j ∈ R (2)The definition of an imputation, Definition 2, needs to be modified in an obvious way to distin-guish the profit shares of buyers and sellers; we will denote an imputation by ( v , u ) . Theorem 3. (Shapley and Shubik [SS71]) The imputation ( v , u ) is in the core of the assignment game ifand only if it is an optimal solution to the dual LP, (2). ∑ ( i , j ) ∈ E w ij x ij s.t. ∑ ( i , j ) ∈ E x ij ≤ ∀ i ∈ V , ∑ ( i , j ) ∈ S x ij ≤ ( | S | − ) ∀ S ⊆ V , S odd, x ij ≥ ∀ ( i , j ) ∈ E (3)The dual of this LP has, in addition to variables corresponding to vertices, v i , exponentially manymore variables corresponding to odd sets, z S , as given in (4). As a result, the entire worth of thegame does not reside on vertices only — it also resides on odd sets.min ∑ i ∈ V v i + ∑ S ⊆ V , odd z S s.t. v i + v j + ∑ S ∋ i , j z S ≥ w ij ∀ ( i , j ) ∈ E , v i ≥ ∀ i ∈ V , z S ≥ ∀ S ⊆ V , S odd (4)There is no natural way of dividing z S among the vertices in S to restore core properties. Thesituation is more serious than that: it turns out that in general, the core of a non-bipartite gamemay be empty.A (folklore) proof of the last fact goes as follows: Consider the graph K , i.e., a clique on threevertices, i , j , k , with a weight of 1 on each edge. Any maximum matching in K has only oneedge, and therefore the worth of this game is 1. Suppose there is an imputation v which lies inthe core. Consider all three two-agent coalitions. Then, we must have: v ( i ) + v ( j ) ≥ v ( j ) + v ( k ) ≥ v ( i ) + v ( k ) ≥ v ( i ) + v ( j ) + v ( k ) ≥ v ( i ) = v ( j ) = v ( k ) = Remark 4.
In the assignment game, monetary transfers are required only between a buyer-sellerpair who are involved in a trade. In contrast, observe that in the approximate core allocationgiven above for K , transfers are even made from agents who make trade to agents who don’t,thereby exploiting the TU market’s capabilities more completely.4 efinition 5. Let p : 2 V → R + be the characteristic function of a game and let 1 ≥ α >
0. Animputation v : V → R + is said to be in the α -approximate core of the game if:1. The total profit allocated by v is at most the worth of the game, i.e., ∑ i ∈ V v ( i ) ≤ p ( V ) .2. The total profit accrued by agents in a sub-coalition S ⊆ V is at least α fraction of the profitwhich S can generate by itself, i.e., ∀ S ⊆ V : ∑ i ∈ S v i ≥ α · p ( S ) .If imputation v is in the α -approximate core of a game, then the ratio of the total profit of anysub-coalition on seceding from the grand coalition to its profit while in the grand coalition isbounded by a factor of at most α .In Section 3 we will need the following notion. Definition 6.
Consider a linear programming relaxation for a maximization problem. For aninstance I of the problem, let OPT ( I ) denote the weight of an optimal solution to I and letOPT f ( I ) denote the weight of an optimal solution to the LP-relaxation of I . Then, the integralitygap of this LP-relaxation is defined to be: inf I OPT ( I ) OPT f ( I ) . -Approximate Core for the Matching Game We will work with the following LP-relaxation of the maximum weight matching problem, (5).This relaxation always has an integral optimal solution in case G is bipartite, but not in generalgraphs. In the latter, its optimal solution is a maximum weight fractional matching in G .max ∑ ( i , j ) ∈ E w ij x ij s.t. ∑ ( i , j ) ∈ E x ij ≤ ∀ i ∈ V , x ij ≥ ∀ ( i , j ) ∈ E (5)Taking v i to be dual variables for the first constraint of (5), we obtain LP (6). Any feasible solutionto this LP is called a cover of G since for each edge ( i , j ) , v i and v j cover edge ( i , j ) in the sensethat v i + v j ≥ w ij . An optimal solution to this LP is a minimum cover . We will say that v i is the profit of vertex i . 5in ∑ i ∈ V v i s.t. v i + v j ≥ w ij ∀ ( i , j ) ∈ E , v i ≥ ∀ i ∈ V (6)By the LP Duality Theorem, the weight of a maximum weight fractional matching equals thetotal profit of a minimum cover. If for graph G , LP (5) has an integral optimal solution, then it iseasy to see that an optimal dual solution gives a way of allocating the total worth which lies inthe core. Deng et. al. [DIN97] prove that the core of this game is non-empty if and only if LP (5)has an integral optimal solution.We will say that a solution x to LP (5) is half-integral if for each edge ( i , j ) , x ij is 0, 1/2 or 1.Balinski [Bal65] showed that the vertices of the polytope defined by the constraints of LP (5)are half-integral, see Theorem 14 below. As a consequence, any optimal solution to LP (5) ishalf-integral. Biro et. al [BKP12] gave an efficient algorithm for finding an optimal half-integralmatching by using an idea of Nemhauser and Trotter [NT75] of doubling edges, hence obtainingan efficient algorithm for determining if the core of the game is non-empty.Our mechanism starts by using the doubling idea of [NT75]. Transform G = ( V , E ) with edge-weights w to graph G ′ = ( V ′ , E ′ ) and edge weights w ′ as follows. Corresponding to each i ∈ V , V ′ has vertices i ′ and i ′′ , and corresponding to each edge ( i , j ) ∈ E , E ′ has edges ( i ′ , j ′′ ) and ( i ′′ , j ′ ) each having a weight of w ij /2.Since each cycle of length k in G is transformed to a cycle of length 2 k in G ′ , the latter graphis bipartite. A maximum weight matching and a minimum cover for G ′ can be computed inpolynomial time [LP86], say x ′ and v ′ , respectively. Next, let x ij = · ( x i ′ , j ′′ + x i ′′ , j ′ ) and v i = ( v i ′ + v i ′′ ) .Clearly, x is an optimal half-integral matching and v is a cover in G . It is easy to see that theweight of x equals the value of v , thereby implying that v is an optimal cover.Edges that are set to half in x form connected components which are either paths or cycles. Forany such path, consider the two matchings obtained by picking alternate edges. The half-integralsolution for this path is a convex combination of these two integral matchings. Therefore boththese matchings must be of equal weight, since otherwise we can obtain a heavier matching. Pickany of them. Similarly, if a cycle is of even length, pick alternate edges and match them. Thistransforms x to a maximum weight half-integral matching in which all edges that are set to halfform disjoint odd cycles. Henceforth we will assume that x satisfies this property.Let C be a half-integral odd cycle in x of length 2 k +
1, with consecutive vertices i , . . . i k + . Let w C = w i , i + w i , i + . . . + w i k + , i and v C = v i + . . . v i k + . On removing any one vertex, say i j ,with its two edges from C , we are left with a path of length 2 k −
1. Let M j be the matchingconsisting of the k alternate edges of this path and let w ( M j ) be the weight of this matching. Lemma 7.
Odd cycle C satisfies: . w C = · v C
2. C has a unique cover: v i j = v C − w ( M j ) , for ≤ j ≤ k + .Proof. We will use the fact that x and v are optimal solutions to LPs (5) and (6), respectively.By the primal complementary slackness condition, for 1 ≤ j ≤ k + w i j , i j + = v i j + v i j + , whereaddition in the subindices is done modulo 2 k +
1; this follows from the fact that x i j , i j + > C we get w C = · v C . By the equalities established in the first part, we get that for 1 ≤ j ≤ k + v C = v i j + w ( M j ) .Rearranging terms gives the lemma.Let M ′ be heaviest matching among M j , for 1 ≤ j ≤ k + Lemma 8. w ( M ′ ) ≥ k k + · v C Proof.
Adding the equality established in the second part of Lemma 7 for all 2 k + j we get: k + ∑ j = w ( M j ) = ( k ) · v C Since M ′ is the heaviest of the 2 k + x to obtain an integral matching T in G as follows. First pickall edges ( i , j ) such that x ij = T . Next, for each odd cycle C , find the heaviest matching M ′ as described above and pick all its edges. Definition 9.
Let 1 > α >
0. A function c : V → R + is said to be an α -approximate cover for G if ∀ ( i , j ) ∈ E : c i + c j ≥ α · w ij Define function f : V → [ , 1 ] as follows: ∀ i ∈ V : f ( i ) = ( k k + if i is in a half-integral cycle of length 2 k + i is not in a half-integral cycle.Next, modify cover v to obtain an approximate cover c as follows: ∀ i ∈ V : c i = f ( i ) · v i . Lemma 10. c is a -approximate cover for G.Proof. Consider edge ( i , j ) ∈ E . Then c i + c j = f ( i ) · v i + f ( j ) · v j ≥ · ( v i + v j ) ≥ · w ij ,where the first inequality follows from the fact that ∀ i ∈ V , f ( i ) ≥ and the second followsfrom the fact that v is a cover for G . 7 echanism 11. ( -Approximate Core Imputation)
1. Compute x and v , optimal solutions to LPs (5) and (6), where x is half-integral.2. Modify x so all half-integral edges form odd cycles.3. ∀ i ∈ V , compute: f ( i ) = ( k k + if i is in a half-integral cycle of length 2 k + ∀ i ∈ V : c i ← f ( i ) · v i .Output c .The mechanism for obtaining imputation c is summarized as Mechanism 11. Theorem 12.
The imputation c is in the -approximate core of the general graph matching game.Proof. We need to show that c satisfies the two conditions given in Definition 5, for α = . By Lemma 8, the weight of the matched edges picked in T from a half-integral odd cycle C of length 2 k + ≥ f ( k ) · v C = ∑ i ∈ C c ( i ) . Next remove all half-integral odd cycles from G toobtain G ′ . Let x ′ and v ′ be the projections of x and v to G ′ .By the first part of Lemma 7, the total decrease in weight in going from x to x ′ equals the totaldecrease in value in going from v to v ′ . Therefore, the weight of x ′ equals the total value of v ′ .Finally, observe that in G ′ , T picks an edge ( i , j ) if and only if x ′ ij = ∀ i ∈ G ′ , c i = v ′ i .Adding the weight of the matching and the value of the imputation c over G ′ and all half-integralodd cycles we get w ( T ) ≥ ∑ i ∈ V c i . Consider a coalition S ⊆ V . Then p ( S ) is the weight of a maximum weight matching in G restricted to S . Assume this matching is ( i , j ) , . . . ( i k , j k ) , where i , . . . i k and j , . . . j k ∈ S . Then p ( S ) = ( w i j + . . . + w i k j k ) . By Lemma 10, c i l + c j l ≥ · w i l , j l , for 1 ≤ l ≤ k .Adding all k terms we get: ∑ i ∈ S c i ≥ · p ( S ) .Observe that for the purpose of Lemma 10, we could have defined f simply as ∀ i ∈ V , f ( i ) = .However in general, this would have left a good fraction of the worth of the game unallocated.8he definition of f given above improves the allocation for agents who are in large odd cyclesand those who are not in odd cycles with respect to matching x . As a result, the gain of a typicalsub-coalition on seceding will be less than a factor of , giving it less incentive to secede. Oneway of formally stating an improved factor is given in Proposition 13; its proof is obvious fromthat of Theorem 12. Proposition 13.
Assume that the underlying graph G has no odd cycles of length less than k + . Thenimputation c is in the k k + -approximate core of the matching game for G. An easy corollary of Balinski’s result, Theorem 14, is that the integrality gap of LP-relaxation (5)is precisely . For completeness, we have provided a proof in Corollary 15. As a consequenceof this fact, improving the approximation factor of an imputation for the matching game is notpossible. Theorem 14. (Balinski [Bal65]) The vertices of the polytope defined by the constraints of LP (5) arehalf-integral.
Corollary 15.
The integrality gap of LP-relaxation (5) is .Proof. From the proof of the first part of Theorem 12 we get: w ( T ) ≥ ∑ i ∈ V c i ≥ · ∑ i ∈ V v i = · w ( x ) .Therefore for any instance I = ( G , w ) , OPT ( I ) OPT f ( I ) ≥
23 .This places a lower bound of the integrality gap of LP-relaxation (5).To place an upper bound of , consider the following infinite family of graphs. For each n , thegraph G n has 6 n vertices i l , j l , k l , for 1 ≤ l ≤ n , and 6 n edges ( i l , j l ) , ( j l , k l ) , ( i l , k l ) , for 1 ≤ l ≤ n all of weight 1. Clearly, OPT ( G n ) = n and OPT f ( G n ) = n . In case a connected graph is desired,add a clique on the 2 n vertices i l , for 1 ≤ l ≤ n , with the weight of each edge being ǫ , where ǫ tends to zero. In the 2/3-approximate core imputation, observe that in an odd cycle of length 2 k + k pairsof agents are matched and one agent is left unmatched. As a consequence, monetary transfersmay be needed from all 2 k matched agents to the unmatched agent. What happens if monetarytransfers to an agent are allowed from only a limited number of other agents? If so, what is thebest approximation factor possible? See also Remark 4.9or the assignment game, Shapley and Shubik are able to characterize “antipodal” points inthe core, i.e., imputations which are maximally distant. An analogous understanding of the -approximate core of the general graph matching game will be desirable. I wish to thank Federico Echenique and Thorben Trobst for valuable discussions.
References [Bal65] Michel Louis Balinski. Integer programming: methods, uses, computations.
Manage-ment science , 12(3):253–313, 1965.[BKP12] P´eter Bir ´o, Walter Kern, and Dani¨el Paulusma. Computing solutions for matchinggames.
International journal of game theory , 41(1):75–90, 2012.[DIN97] Xiaotie Deng, Toshihide Ibaraki, and Hiroshi Nagamochi. Algorithms and complexityin combinatorial optimization games. In
Proc. 8th ACM Symp. on Discrete Algorithms ,1997.[EK01] Kimmo Eriksson and Johan Karlander. Stable outcomes of the roommate game withtransferable utility.
International Journal of Game Theory , 29(4):555–569, 2001.[LP86] L. Lov´asz and M.D. Plummer.
Matching Theory . North-Holland, Amsterdam–New York,1986.[Mou14] Herv´e Moulin.
Cooperative microeconomics: a game-theoretic introduction , volume 313.Princeton University Press, 2014.[NT75] George L Nemhauser and Leslie Earl Trotter. Vertex packings: structural properties andalgorithms.
Mathematical Programming , 8(1):232–248, 1975.[SS71] Lloyd S Shapley and Martin Shubik. The assignment game I: The core.