Matroid matching: The power of local search

We consider the classical matroid matching problem. Unweighted matroid matching for linearly represented matroids was solved by Lovász, and the problem is known to be intractable for general matroids. We present a polynomial-time approximation scheme for unweighted matroid matching for general matroids. In contrast, we show that natural linear-programming relaxations that have been studied have an Ω(n) integrality gap, and, moreover, Ω(n) rounds of the Sherali- Adams hierarchy are necessary to bring the gap down to a constant. More generally, for any fixed k ≥ 2 and ε > 0, we obtain a (k/2+ε)-approximation for matroid matching in k-uniform hypergraphs, also known as the matroid k-parity problem. As a consequence, we obtain a (k/2+ε)-approximation for the problem of finding the maximum-cardinality set in the intersection of k matroids. We also give a 3/2-approximation for the weighted version of a special case of matroid matching, the matchoid problem. © 2013 Society for Industrial and Applied Mathematics.