Result: Matroid Matching: the Power of Local Search (Extended Abstract)

We consider the classical matroid matching problem. Unweighted matroid matching for linear matroids was solved by Lovász, and the problem is known to be intractable for general matroids. We present a PTAS for unweighted matroid matching for general matroids. In contrast, we show that natural LP relaxations 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 ɛ> 0, we obtain a (k/2 + ɛ)-approximation for matroid matching in k-uniform hypergraphs, also known as the matroid k-parity problem. As a consequence, we obtain a (k/2 + ɛ)-approximation for the problem of finding the maximum-cardinality set in the intersection of k matroids. We have also designed a 3/2approximation for the weighted version of a special case of matroid matching, the matchoid problem.