MATROID MATCHING: THE POWER OF LOCAL SEARCH
We consider the classical matroid matching problem. Unweighted matroid matching for linearly represented matroids was solved by Lovasz, 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 Omega(n) integrality gap, and, moreover, Omega(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 epsilon > 0, we obtain a (k/2+ epsilon)-approximation for matroid matching in k-uniform hypergraphs, also known as the matroid k-parity problem. As a consequence, we obtain a (k/2+ epsilon)-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.
matroid matching local search sherali-adams hierarchy chvatal closure