The Minimum Feasible Tileset Problem

  • Yann Disser
  • Stefan Kratsch
  • Manuel Sorge
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8952)


We consider the Minimum Feasible Tileset problem: Given a set of symbols and subsets of these symbols (scenarios), find a smallest possible number of pairs of symbols (tiles) such that each scenario can be formed by selecting at most one symbol from each tile. We show that this problem is \(\mathsf {NP}\)-complete even if each scenario contains at most three symbols. Our main result is a 4/3-approximation algorithm for the general case. In addition, we show that the Minimum Feasible Tileset problem is fixed-parameter tractable both when parameterized with the number of scenarios and with the number of symbols.


  1. 1.
    Bansal, N., Caprara, A., Sviridenko, M.: A new approximation method for set covering problems, with applications to multidimensional bin packing. SIAM J. Comput. 39(4), 1256–1278 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Biedl, T., Chan, T., Ganjali, Y., Hajiaghayi, M., Wood, D.: Balanced vertex-orderings of graphs. Discrete Appl. Math. 148(1), 27–48 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Buchin, K., van Kreveld, M.J., Meijer, H., Speckmann, B., Verbeek, K.: On planar supports for hypergraphs. J. Graph Algorithms Appl. 15(4), 533–549 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Chen, J., Komusiewicz, C., Niedermeier, R., Sorge, M., Suchý, O., Weller, M.: Effective and efficient data reduction for the subset interconnection design problem. In: Cai, L., Cheng, S.-W., Lam, T.-W. (eds.) ISAAC 2013. LNCS, vol. 8283, pp. 361–371. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  5. 5.
    Cygan, M.: Improved approximation for 3-dimensional matching via bounded pathwidth local search. In: Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 509–518 (2013)Google Scholar
  6. 6.
    Dell, H., Marx, D.: Kernelization of packing problems. In: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 68–81 (2012)Google Scholar
  7. 7.
    Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. In: Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC), pp. 251–260 (2010)Google Scholar
  8. 8.
    Disser, Y., Matuschke, J.: Degree-constrained orientations of embedded graphs. In: Chao, K.-M., Hsu, T., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 506–516. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  9. 9.
    Du, D.-Z., Miller, Z.: Matroids and subset interconnection design. SIAM J. Discrete Math. 1(4), 416–424 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Frank, A., Gyárfás, A.: How to orient the edges of a graph. Colloquia mathematica societatis Janos Bolyai 18, 353–364 (1976)Google Scholar
  11. 11.
    Frank, A., Tardos, É.: An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica 7(1), 49–65 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of NP-Completeness. W.H.Freeman and Company, New York (1979) zbMATHGoogle Scholar
  13. 13.
    Gottlob, G., Greco, G.: On the complexity of combinatorial auctions: structured item graphs and hypertree decomposition. In: Proceedings of the 8th ACM Conference on Electronic Commerce (EC), pp. 152–161 (2007)Google Scholar
  14. 14.
    Hakimi, S.: On the degrees of the vertices of a directed graph. J. Frankl. Inst. 279(4), 290–308 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Johnson, D.S., Pollak, H.O.: Hypergraph planarity and the complexity of drawing Venn diagrams. J. Graph Theory 11(3), 309–325 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Kannan, R.: Minkowski’s convex body theorem and integer programming. Math. Oper. Res. 12, 415–440 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Koutis, I.: Faster algebraic algorithms for path and packing problems. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 575–586. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  18. 18.
    Lenstra, H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8, 538–548 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Sviridenko, M., Ward, J.: Large neighborhood local search for the maximum set packing problem. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part I. LNCS, vol. 7965, pp. 792–803. Springer, Heidelberg (2013) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Technische Universität BerlinBerlinGermany

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