The complexity of maximum matroid-greedoid intersection and weighted greedoid maximization

Authors:
Taneli Mielikäinen;Esko Ukkonen
Affiliations:
Department of Computer Science, P.O. Box 68, FIN-00014 University of Helsinki, Finland;Department of Computer Science, P.O. Box 68, FIN-00014 University of Helsinki, Finland
Venue:
Discrete Applied Mathematics - Special issue: Efficient algorithms
Year:
2006

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Abstract

The maximum intersection problem for a matroid and a greedoid, given by polynomial-time oracles, is shown NP-hard by expressing the satisfiability of boolean formulas in 3-conjunctive normal form as such an intersection. The corresponding approximation problems are shown NP-hard for certain approximation performance bounds. Moreover, some natural parameterized variants of the problem are shown W[P]-hard. The results are in contrast with the maximum matroid-matroid intersection which is solvable in polynomial time by an old result of Edmonds. We also prove that it is NP-hard to approximate the weighted greedoid maximization within 2^n^^^O^^^(^^^1^^^) where n is the size of the domain of the greedoid.