Better Approximation for Weighted $k$-Matroid Intersection

TL;DR

This paper introduces a new approximation algorithm for the weighted $k$-matroid intersection problem, improving the approximation guarantee and leveraging randomized reductions and insights from unweighted cases.

Contribution

It provides the first improvement over the longstanding approximation guarantee for weighted $k$-matroid intersection and introduces a novel randomized reduction approach.

Findings

Achieves a $(k+1)/(2 imes ext{ln} 2)$-approximation ratio.

First improvement over greedy for weighted matroid $k$-parity.

Utilizes randomized iterative solving of unweighted instances.

Abstract

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 $(k+1)/(2 \ln 2)$-approximation algorithm for the weighted $k$-matroid intersection problem. This is the first improvement over the longstanding $(k-1)$-guarantee of Lee, Sviridenko and Vondr\'ak (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 Vondr\'ak have designed a $k/2$-approximation algorithm. We analyze this procedure by constructing refined matroid exchanges and leveraging randomness to avoid bad local minima.