A Parameterized View on Matroid Optimization Problems

Matroid theory gives us powerful techniques for understanding combinatorial optimization problems and for designing polynomialtime algorithms. However, several natural matroid problems, such as 3-matroid intersection, are NP-hard. Here we investigate these problems from the parameterized complexity point of view: instead of the trivial O(n) time brute force algorithm for finding a k-element solution, we try to give algorithms with uniformly polynomial (i.e., f(k) · n) running time. The main result is that if the ground set of a represented matroid is partitioned into blocks of size l, then we can determine in f(k, l)·n randomized time whether there is an independent set that is the union of k blocks. As consequence, algorithms with similar running time are obtained for other problems such as finding a k-set in the intersection of l matroids, or finding k terminals in a network such that each of them can be connected simultaneously to the source by l disjoint paths.