Abstract
Gabow and Tarjan [9] provided a very elegant and fast algorithm for the following problem: given a matroid defined on a red and blue colored ground set, determine a basis of minimum cost among those with k red elements, or decide that no such basis exists. In this paper, we investigate possible extensions of this result from ordinary matroids to the more general notion of poset matroids. Poset matroids (also called distributive supermatroids) are defined on the collection of all ideals of an underlying partial order on the ground set. We show that the problem on general poset matroids becomes NP-hard, already if the underlying poset consists of binary trees of height two. On the positive side, we present two polynomial algorithms: one for integer polymatroids, i.e., the case where the poset consists of disjoint chains, and one for the problem to determine a minimum cost ideal of size l with k red elements, i.e., the uniform rank-l poset matroid, on series-parallel posets.
The authors thank the German Research Association (DFG) for funding this work (Research Grants SFB 666 and PE 1434/3-1).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Aggarwal, V., Aneja, Y.P., Nair, K.: Minimal spanning tree subject to a side constraint. Comput. Oper. Res. 9(4), 287–296 (1982)
Barnabei, M., Nicoletti, G., Pezzoli, L.: The symmetric exchange property for poset matroids. Adv. Math. 102, 230–239 (1993)
Dunstan, F.D.J., Ingleton, A.W., Welsh, D.J.A.: Supermatroids. In: Welsh, D.J.A., Woodall, D.R. (eds.) Combinatorics (Proceedings of the Conference on Combinatorial Mathematics), pp. 72–122. The Institute of Mathematics and its Applications (1972)
Faigle, U.: The greedy algorithm for partially ordered sets. Discrete Math. 28, 153–159 (1979)
Faigle, U.: Matroids on ordered sets and the greedy algorithm. Ann. Discrete Math. 19, 115–128 (1984)
Faigle, U., Kern, W.: Computational complexity of some maximum average weight problems with precedence constraints. Oper. Res. 42(4), 688–693 (1994). http://dx.doi.org/10.1287/opre.42.4.688
Fleiner, T., Frank, A., Iwata, S.: A constrained independent set problem for matroids. Technical report TR-2003-01, Egerváry Research Group, Budapest (2003). www.cs.elte.hu/egres
Fujishige, S.: Submodular Functions and Optimization. Annals of Discrete Mathematics, vol. 58, 2nd edn. Elsevier, Amsterdam (2005)
Gabow, H.N., Tarjan, R.E.: Efficient algorithms for a family of matroid intersection problems. J. Algorithms 5(1), 80–131 (1984)
Groenevelt, H.: Two algorithms for maximizing a separable concave function over a polymatroid feasible region. Eur. J. Oper. Res. 54, 227–236 (1991)
Lüthen, H.: On matroids and shortest path with additional precedence constraints. Master’s thesis, Technische Universität Berlin (2012)
Martins, E.: On a multicriteria shortest path problem. Eur. J. Oper. Res. 16, 236–245 (1984)
Mulmuley, K., Vazirani, U.V., Vazirani, V.V.: Matching is as easy as matrix inversion. Combinatorica 7(1), 105–113 (1987)
Murota, K.: Discrete Convex Analysis. Soc. Ind. Appl. Math. (SIAM) (2003)
Räbiger, D.: Semi-Präemptives transportieren [in German]. Ph.D. thesis, Universität zu Köln (2005)
Valdes, J., Tarjan, R.E., Lawler, E.L.: The recognition of series parallel digraphs. In: Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, STOC 1979, New York, NY, USA, pp. 1–12. ACM (1979)
Yuster, R.: Almost exact matchings. Algorithmica 63, 39–50 (2011)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Gottschalk, C., Lüthen, H., Peis, B., Wierz, A. (2016). Optimization Problems with Color-Induced Budget Constraints. In: Cerulli, R., Fujishige, S., Mahjoub, A. (eds) Combinatorial Optimization. ISCO 2016. Lecture Notes in Computer Science(), vol 9849. Springer, Cham. https://doi.org/10.1007/978-3-319-45587-7_17
Download citation
DOI: https://doi.org/10.1007/978-3-319-45587-7_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-45586-0
Online ISBN: 978-3-319-45587-7
eBook Packages: Computer ScienceComputer Science (R0)