Exact and approximation algorithms for weighted matroid intersection

Abstract

In this paper, we propose new exact and approximation algorithms for the weighted matroid intersection problem. Our exact algorithm is faster than previous algorithms when the largest weight is relatively small. Our approximation algorithm delivers a $(1-\epsilon )$-approximate solution with a running time significantly faster than most known exact algorithms. The core of our algorithms is a decomposition technique: we decompose an instance of the weighted matroid intersection problem into a set of instances of the unweighted matroid intersection problem. The computational advantage of this approach is that we can make use of fast unweighted matroid intersection algorithms as a black box for designing algorithms. More precisely, we show that we can solve the weighted matroid intersection problem via solving W instances of the unweighted matroid intersection problem, where W is the largest given weight, assuming that all given weights are integral. Furthermore, we can find a $(1-\epsilon )$-approximate solution via solving $O(\epsilon ^{-1} \log r)$ instances of the unweighted matroid intersection problem, where r is the smaller rank of the two given matroids. Our algorithms make use of the weight-splitting approach of Frank (J Algorithms 2(4):328–336, 1981) and the geometric scaling scheme of Duan and Pettie (J ACM 61(1):1, 2014). Our algorithms are simple and flexible: they can be adapted to special cases of the weighted matroid intersection problem, using specialized unweighted matroid intersection algorithms. In addition, we give a further application of our decomposition technique: we solve efficiently the rank-maximal matroid intersection problem, a problem motivated by matching problems under preferences.