Fractional matroid matchings

Abstract

This paper introduces a linear relaxation of the matroid matching problem, called the fractional matroid matching problem. When the matroid matching problem is in fact a matroid intersection problem, the fractional matroid matching polytope and the matroid matching polytope coincide. When the matroid matching problem is in fact a matching problem in a graph, the fractional matroid matching polytope and the classic fractional matching polytope of the graph coincide. Thus, the fractional matroid matching polytope may properly contain the matroid matching polytope. The fractional matroid matching polytope is a lattice polyhedron and, although its extreme points are not all integral, they are half-integral. This paper establishes strong relationships between extreme points of the fractional matroid matching polytope and those of graphic fractional matching polyhedra. Despite these strong relationships, adding the rank 1 inequalities does not define the matroid matching polytope. In fact, even the matching polytope of a graphic matroid is not generally described by adding these inequalities.