A parameterized view on matroid optimization problems |
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Authors: | Dániel Marx |
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Affiliation: | Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Budapest H-1521, Hungary |
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Abstract: | Matroid theory gives us powerful techniques for understanding combinatorial optimization problems and for designing polynomial-time 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 nO(k) time brute force algorithm for finding a k-element solution, we try to give algorithms with uniformly polynomial (i.e., f(k)⋅nO(1)) running time. The main result is that if the ground set of a represented linear matroid is partitioned into blocks of size ?, then we can determine in randomized time f(k,?)⋅nO(1) whether there is an independent set that is the union of k blocks. As a consequence, algorithms with similar running time are obtained for other problems such as finding a k-element set in the intersection of ? matroids, or finding k terminals in a network such that each of them can be connected simultaneously to the source by ? disjoint paths. |
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Keywords: | Matroids Fixed-parameter tractability Combinatorial optmization |
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