A compact linear program for testing optimality of perfect matchings Paolo Ventura Istituto di Analisi dei Sistemi ed Informatica Viale Manzoni, 30 00185 Roma Italy ventura@iasi.rm.cnr.it Friedrich Eisenbrand Max-Planck-Institut für Informatik Stuhlsatzenhausweg 85 66123 Saarbrücken Germany eisen@mpi-sb.mpg.de May 14, 2003 Abstract It is a longstanding open problem whether there exists a polynomial size description of the perfect matching polytope. We give a partial answer to this question by prov- ing the following result. The polyhedron defined by the constraints of the perfect matching polytope which are active at a given perfect matching can be obtained as the projection of a compact polyhedron. Thus there exists a compact linear program which is unbounded if and only if the perfect matching is not optimal with respect to a given edge weight. This result provides a simple reduction of the maximum weight perfect matching problem to compact linear programming. Keywords: Matching, linear programming compact linear programming. 1 Introduction A perfect matching of a graph G VE is a set M of edges, such that each node of G is incident with exactly one edge in M. A central problem in algorithms is to find a per- fect matching M of G with maximal weight cM ∑ e M c e with respect to given edge weights c E . This problem was solved by Edmonds [4, 3] with his blossom algorithm. The characteristic vector χ M 01 E of a perfect matching M is a 0/1 vector which in- dicates whether an edge e E is a member of the matching or not via the value of its components, i.e., χ M e 1 if e M and χ M e 0 otherwise. The convex hull of the charac- teristic vectors of perfect matchings of G is called the perfect matching polytope P G of G. Edmonds showed that PG is described by the following set of linear inequalities. Corresponding author 1