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We show that any k-regular bipartite graph with 2n vertices has at least ( (k\Gamma1) k\Gamma1 k k\Gamma2 ) n perfect matchings (1-factors). 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.

Petersen's theorem is a classic result in matching theory from 1891, stating that every 3-regular bridgeless graph has a perfect matching. Our work explores efficient algorithms for finding perfect matchings in such graphs. Previously, the only relevant matching algorithms were for general graphs, and the fastest algorithm ran in O(n^3/2) time for 3-regular graphs. We have developed an O(n log^4 n)-time algorithm for perfect matching in a 3-regular bridgeless graph. When the graph is also planar, we have as the main result of our paper an optimal O(n)-time algorithm. We present three applications of this result: terrain guarding, adaptive mesh refinement, and quadrangulation.

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 min-max 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 f-factor-theorem and Edmonds' matroid intersection theorem. 1 Introduction Two of the most well-known 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...

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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 Tutte-Berge 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]

No complete characterization of rank facet producing graphs is known. In 1977, Balas and Zemel gave a necessary for a graph to be rank-facet producing, and in 1975, Chvtal gave a sufficient condition, which motivated the study of the class of α-critical graphs. We give two strengthening of Balas and Zemel's necessary condition, and one of Chvatal's sufficient condition.

We analyze an infinite horizon, non-cooperative 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 take-it-or-leave-it 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: Non-cooperative 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,