FACES OF MATCHING POLYHEDRA

Let G = (V, E, ~) be a finite loopless graph, let \nb=(bi:ieV) be a vector of positive integers. A \nfeasible matching is a vector X = (x.: j e: E) \nJ \nof nonnegative \nintegers such that for each node i of G, the sum of the \nover the edges j of G incident with i is no \ngreater than bi. The matching polyhedron P(G, b) is the \nconvex hull of the set of feasible matchings. \nIn Chapter 3 we describe a version of Edmonds' blossom \nalgorithm which solves the problem of maximizing C • X \nover P (G, b) where c =. (c.: j e: E) \nJ \nis an arbitrary real \nvector. This algorithm proves a theorem of Edmonds which \ngives a set of linear inequalities sufficient to define \nP(G, b). \nIn Chapter 4 we prescribe the unique subset of these \ninequalities which are necessary to define P(G, b), that \nis, we characterize the facets of P(G, b). We also \ncharacterize the vertices of P(G, b), thus describing the \nstructure possessed by the members of the minimal set X \nof feasible matchings of G such that for any real vector \nc = (c.: j e: E), c • x is maximized over P(G, b) \nJ \nmember of X. \nby a \nIn Chapter 5 we present a generalization of the blossom \nalgorithm which solves the problem: maximize c • x over \na face F of P(G, b) for any real vector c = (c.: j e: E). \nJ \nIn other words, we find a feasible matching x of G which \nsatisfies the constraints obtained by replacing an arbitrary \nsubset of the inequalities which define P(G, b) by equations and which maximizes c • x subject to this \nrestriction. We also describe an application of this \nalgorithm to matching problems having a hierarchy of objective \nfunctions, so called ''multi-optimization'' problems. \nIn Chapter 6 we show how the blossom algorithm can be \ncombined with relatively simple initialization algorithms \nto give an algorithm which solves the following postoptimality \nproblem. Given that we know a matching 0 x £ P(G, b) \nmaximizes c · x over P(G, b), we wish to utilize 0 \nX \nwhich \nto \nfind a feasible matching x' £ P(G, b') which maximizes \nc • x over P(G, b'), where b' = (b!: i £ V) \n]_ \nvector of positive integers and \narbitrary real vector. \nc=(c.:j£E) \nJ \nis a \nis an \nIn Chapter 7 we describe a computer implementation of \nthe blossom algorithm described herein.

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