1 Matching using Linear Programming We look at the linear programming method for the maximum matching and perfect matching problems. Given a graph G (V, E), an integer linear program (ILP) for the maximum matching problem can be written by dening a variable x e for each edge e E and a constraint for each vertex u V as follows: Maximize X eE xTheorem 4. 1 For a given bipartite graph G, a matching M is maximum if and only if G has no augmenting paths with respect to M. Proof: ()) We prove this by contrapositive, i. e. , by showing that if G has an augmenting path, then maximum bipartite matching linear programming

1. Lecture notes on bipartite matching February 9th, 2009 2 1. 1 Maximum cardinality matching problem Before describing an algorithm for solving the maximum cardinality matching problem, one would like to be able to prove optimality of a matching (without reference to any algorithm).

1 Bipartite matching and vertex covers Recall that a bipartite graph G (V; E) is a graph whose vertices can be divided into two disjoint sets such that every edge connects one node in one set to a node in the other. De nition 1 (Matching, vertex cover). A matching is a disjoint subset of edges, i. e. , a subset of edges that do not share a common vertex. Konig's theorem says that, in a bipartite graph, the size of the maximum matching equals the size of the minimum vertex cover. This theorem has several proofs; I would like to know if the following proof, based on primaldual linear program, is valid. This is mainly an exercise in**maximum bipartite matching linear programming** Sep 14, 2009 Bipartite matchings via linear programming An important tool for combinatorial optimisation problems, and graph problems among them, is linear programming, a method (and a problem) to find the optimum of a linear function on a polyhedron (i. e. on an intersection of halfspaces. )

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Maximum Bipartite Matching set A; set B; set E within A cross B; # a bipartite graph var X e in E 0, 1; # variable for each edge maximize numedges: sum (u, v) in E X[u, v; s. t. matchA u in A: sum (u, v) in E X[u, v 1; s. t. matchB v in B: sum (u, v) in E X[u, v 1; end; *maximum bipartite matching linear programming* polytope, and can be handled with linear programming methods by the total unimodularity of the incidence matrix of a bipartite graph. In this chapter, graphs can be assumed to be simple. 18. 1. The matching and the perfect matching polytope Let G (V, E) be a graph. The perfect matching polytope Pperfect matching(G) If the graph is not bipartite, then the above scheme does not work since we might have oddlength cycles. QUESTION: Is there a similar algorithm for finding a maximumcardinality matching, even in unweighted graphs, based on rounding a fractional solution to the above LP? Maximum Bipartite Matching A matching in a Bipartite Graph is a set of the edges chosen in such a way that no two edges share an endpoint. A maximum matching is a matching of maximum size (maximum number of edges).