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Counting 1Factors in Regular Bipartite Graphs
, 1998
"... We show that any kregular bipartite graph with 2n vertices has at least ( (k\Gamma1) k\Gamma1 k k\Gamma2 ) n perfect matchings (1factors). Equivalently, this is a lower bound on the permanent of any nonnegative integer n \Theta n matrix with each row and column sum equal to k. For any k, the ..."
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We show that any kregular bipartite graph with 2n vertices has at least ( (k\Gamma1) k\Gamma1 k k\Gamma2 ) n perfect matchings (1factors). Equivalently, this is a lower bound on the permanent of any nonnegative integer n \Theta n matrix with each row and column sum equal to k. For any k, the base (k\Gamma1) k\Gamma1 k k\Gamma2 is largest possible. 1.
The Membership Problem in Jump Systems
, 1995
"... A jump system is a set of lattice points satisfying a certain exchange axiom. This notion was introduced by Bouchet and Cunningham [2], as a common generalization of (among others) the sets of bases of a matroid and degree sequences of subgraphs of a graph. We prove, under additional assumptions, ..."
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A jump system is a set of lattice points satisfying a certain exchange axiom. This notion was introduced by Bouchet and Cunningham [2], as a common generalization of (among others) the sets of bases of a matroid and degree sequences of subgraphs of a graph. We prove, under additional assumptions, a minmax formula for the distance of a lattice point from a jump system. The conditions are met in the examples above, and so our formula contains, as special cases, Tutte's ffactortheorem and Edmonds' matroid intersection theorem. 1 Introduction Two of the most wellknown and important classes of combinatorial optimization problems are the matching problem and the matroid intersection problem. These two problems have a lot in common: both are polynomial time solvable; the bipartite matching problem is a common special case; and many of the algorithms to solve them use alternating paths. It is therefore a natural idea to find a common generalization of them. Several common generali...
Total Dual Integrality Of Matching Forest Constraints
, 1998
"... this paper we prove that the system given by Giles is totally dual integral (cf. [15]). This means that the linear program of maximizing an integer objective function over the constraints has integer primal and dual solutions. It generalizes the total dual integrality of the matching constraints in ..."
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Cited by 3 (0 self)
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this paper we prove that the system given by Giles is totally dual integral (cf. [15]). This means that the linear program of maximizing an integer objective function over the constraints has integer primal and dual solutions. It generalizes the total dual integrality of the matching constraints in an undirected graph, proved by Cunningham and Marsh [3] (which generalizes the TutteBerge formula for the maximum size of a matching (cf. [13])), and the total dual integrality of the branching constraints in a directed graph, proved implicitly by Edmonds [5], Bock [1], and Fulkerson [8]
About Facets of the Stable Set Polytope of a Graph
, 2000
"... No complete characterization of rank facet producing graphs is known. In 1977, Balas and Zemel gave a necessary for a graph to be rankfacet producing, and in 1975, Chvtal gave a sufficient condition, which motivated the study of the class of αcritical graphs. We give two strengthening of Bal ..."
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No complete characterization of rank facet producing graphs is known. In 1977, Balas and Zemel gave a necessary for a graph to be rankfacet producing, and in 1975, Chvtal gave a sufficient condition, which motivated the study of the class of αcritical graphs. We give two strengthening of Balas and Zemel's necessary condition, and one of Chvatal's sufficient condition.
Coalitional Bargaining in Networks ∗ Thành Nguyen †
, 2011
"... We analyze an infinite horizon, noncooperative bargaining model for TU games with general coalitional structure. In each period of the game an opportunity for a feasible coalition to form arises according to a stochastic process, and a randomly selected agent in the coalition makes a takeitorlea ..."
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We analyze an infinite horizon, noncooperative bargaining model for TU games with general coalitional structure. In each period of the game an opportunity for a feasible coalition to form arises according to a stochastic process, and a randomly selected agent in the coalition makes a takeitorleaveit offer. Agents that reach an agreement exit the game and are replaced by clones. We characterize the set of stationary equilibria by a convex program. We examine the implications of this characterization when the feasible coalitions are determined by an underlying network. We show how an agent’s payoff is related to the centrality of his position in the network. Keywords: Noncooperative Bargaining, Coalition Formation, Network Games. ∗The author thanks Rakesh Vohra for many fruitful discussions and his tremendous help. I also would like to thank (in random order) Randall Berry, Michael Honig, Éva Tardos, Vijay Subramanian,