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Optimization problems with color-induced budget constraints

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Abstract

In 1984, Gabow and Tarjan 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 extensions of this problem from ordinary matroids to the more general notion of poset matroids which take precedence constraints on the ground set into account. We show that the problem on general poset matroids becomes -hard, already if the underlying partially ordered set (poset) consists of binary trees of height two. On the positive side, we present two algorithms: a pseudopolynomial one for integer polymatroids, i.e., the case where the poset consists of disjoint chains, and a polynomial algorithm 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.

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References

  • Aggarwal V, Aneja YP, Nair K (1982) Minimal spanning tree subject to a side constraint. Comput Oper Res 9(4):287–296

    Article  Google Scholar 

  • Barnabei M, Nicoletti G, Pezzoli L (1993) The symmetric exchange property for poset matroids. Adv Math 102:230–239

    Article  MathSciNet  MATH  Google Scholar 

  • Dunstan FDJ, Ingleton AW, Welsh DJA (1972) Supermatroids. In: Welsh DJA, Woodall DR (eds) Combinatorics (proceedings of the conference on combinatorial mathematics). pp. 72–122. The Institute of Mathematics and its Applications

  • Faigle U (1979) The greedy algorithm for partially ordered sets. Discrete Math 28:153–159

    Article  MathSciNet  MATH  Google Scholar 

  • Faigle U (1984) Matroids on ordered sets and the greedy algorithm. Ann Discrete Math 19:115–128

    MathSciNet  MATH  Google Scholar 

  • Faigle U, Kern W (1994) Computational complexity of some maximum average weight problems with precedence constraints. Oper Res 42(4):688–693. doi:10.1287/opre.42.4.688

    Article  MathSciNet  MATH  Google Scholar 

  • Fleiner T, Frank A, Iwata S (2003) A constrained independent set problem for matroids. Technical Report TR-2003-01, Egerváry Research Group, Budapest, www.cs.elte.hu/egres

  • Fujishige S (2005) Submodular functions and optimization, Annals of Discrete Mathematics, vol 58, 2nd edn. Elsevier, Amsterdam

    MATH  Google Scholar 

  • Gabow HN, Tarjan RE (1984) Efficient algorithms for a family of matroid intersection problems. J Algorithms 5(1):80–131

    Article  MathSciNet  MATH  Google Scholar 

  • Groenevelt H (1991) Two algorithms for maximizing a separable concave function over a polymatroid feasible region. J Oper Res 54:227–236

    Article  MATH  Google Scholar 

  • Jensen PM, Korte B (1982) Complexity of matroid property algorithms. SIAM J Comput 11(1):184–190. doi:10.1137/0211014

    Article  MathSciNet  MATH  Google Scholar 

  • Lovász L (1980) Matroid matching and some applications. J Comb Theory Ser B 28(2):208–236

    Article  MathSciNet  MATH  Google Scholar 

  • Lüthen H (2012) On matroids and shortest path with additional precedence constraints. Master’s thesis, Technische Universität Berlin

  • Martins E (1984) On a multicriteria shortest path problem. Eur J Oper Res 16:236–245

    Article  MathSciNet  MATH  Google Scholar 

  • Mulmuley K, Vazirani UV, Vazirani VV (1987) Matching is as easy as matrix inversion. Combinatorica 7(1):105–113

    Article  MathSciNet  MATH  Google Scholar 

  • Murota K (2003) Discrete convex analysis. Society for Industrial and Applied Mathematics (SIAM). doi:10.1137/1.9780898718508

  • Räbiger D (2005) Semi-Präemptives Transportieren [in German]. Ph.D. thesis, Universität zu Köln

  • Valdes J, Tarjan RE, Lawler EL (1979) The recognition of series parallel digraphs. In: Proceedings of the eleventh annual ACM symposium on theory of computing. TOC ’79, ACM, New York, NY, USA, pp 1–12

  • Yuster R (2011) Almost exact matchings. Algorithmica 63:39–50

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank the reviewers for their valuable comments, especially for pointing out that the proof of Theorem 4 could be significantly shortened.

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Authors and Affiliations

  1. RWTH Aachen University, Aachen, Germany

    Corinna Gottschalk, Britta Peis & Andreas Wierz

  2. Siemens Corporate Technology, Munich, Germany

    Corinna Gottschalk

  3. TU Darmstadt, Darmstadt, Germany

    Hendrik Lüthen

Authors
  1. Corinna Gottschalk
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  2. Hendrik Lüthen
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  3. Britta Peis
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  4. Andreas Wierz
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Corresponding author

Correspondence to Hendrik Lüthen.

Additional information

The authors thank the German Research Association (DFG) for funding this work (Research Grants SFB 666 and PE 1434/3-1).

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Gottschalk, C., Lüthen, H., Peis, B. et al. Optimization problems with color-induced budget constraints. J Comb Optim 36, 861–870 (2018). https://doi.org/10.1007/s10878-017-0182-5

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  • DOI: https://doi.org/10.1007/s10878-017-0182-5

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