Efficient Theoretic and Practical Algorithms for Linear Matroid Intersection Problems

Abstract

Efficient algorithms for the matroid intersection problem, both cardinality and weighted versions, are presented. The algorithm for weighted intersection works by scaling the weights. The cardinality algorithm is a special case, but takes advantage of greater structure. Efficiency of the algorithms is illustrated by several implementations on linear matroids. Consider a linear matroid withmelements and rankn. Assume all element weights are integers of magnitude at mostN. Our fastest algorithms use timeO(mn1.77log(nN)) andO(mn1.62) for weighted and unweighted intersection, respectively; this improves the previous best bounds,O(mn2.4) andO(mn2logn), respectively. Corresponding improvements are given for several applications of matroid intersection to numerical computation and dynamic systems.