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 inmaximum 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. )
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).