Better Approximation for Weighted 𝑘-Matroid Intersection

We consider the problem of finding an independent set of maximum weight simultaneously contained in k matroids over a common ground set. This k-matroid intersection problem appears naturally in many contexts, for example in generalizing graph and hypergraph matching problems. In this paper, we provide a (k+1)/(2 ln2)-approximation algorithm for the weighted k-matroid intersection problem. This is the first improvement over the longstanding (k−1)-guarantee of Lee, Sviridenko and Vondrák (2009). Along the way, we also give the first improvement over greedy for the more general weighted matroid k-parity problem. Our key innovation lies in a randomized reduction in which we solve almost unweighted instances iteratively. This perspective allows us to use insights from the unweighted problem for which Lee, Sviridenko, and Vondrák have designed a k/2-approximation algorithm. We analyze this procedure by constructing refined matroid exchanges and leveraging randomness to avoid bad local minima.