Parameterized Resiliency Problems via Integer Linear Programming

  • Jason Crampton
  • Gregory Gutin
  • Martin Koutecký
  • Rémi Watrigant
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10236)


We introduce a framework in parameterized algorithms whose purpose is to solve resiliency versions of decision problems. In resiliency problems, the goal is to decide whether an instance remains positive after any (appropriately defined) perturbation has been applied to it. To tackle these kinds of problems, some of which might be of practical interest, we introduce a notion of resiliency for Integer Linear Programs (ILP) and show how to use a result of Eisenbrand and Shmonin (Math. Oper. Res., 2008) on Parametric Linear Programming to prove that ILP Resiliency is fixed-parameter tractable (FPT) under a certain parameterization.

To demonstrate the utility of our result, we consider natural resiliency version of several concrete problems, and prove that they are FPT under natural parameterizations. Our first result, for a problem which is of interest in access control, subsumes several FPT results and solves an open question from Crampton et al. (AAIM 2016). The second concerns the Closest String problem, for which we extend an FPT result of Gramm et al. (Algorithmica, 2003). We also consider problems in the fields of scheduling and social choice. We believe that many other problems can be tackled by our framework.


  1. 1.
    Asahiro, Y., Jansson, J., Miyano, E., Ono, H., Zenmyo, K.: Approximation algorithms for the graph orientation minimizing the maximum weighted outdegree. In: Kao, M.-Y., Li, X.-Y. (eds.) AAIM 2007. LNCS, vol. 4508, pp. 167–177. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-72870-2_16 CrossRefGoogle Scholar
  2. 2.
    Bredereck, R., Faliszewski, P., Niedermeier, R., Skowron, P., Talmon, N.: Elections with few candidates: prices, weights, and covering problems. In: Walsh, T. (ed.) ADT 2015. LNCS (LNAI), vol. 9346, pp. 414–431. Springer, Cham (2015). doi: 10.1007/978-3-319-23114-3_25 CrossRefGoogle Scholar
  3. 3.
    Chitnis, R., Cygan, M., Hajiaghayi, M., Marx, D.: Directed subset feedback vertex set is fixed-parameter tractable. ACM Trans. Algorithms 11(4), 1–28 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Crampton, J., Gutin, G., Watrigant, R.: An approach to parameterized resiliency problems using integer Linear Programming. CoRR, abs/1605.08738 (2016)Google Scholar
  5. 5.
    Crampton, J., Gutin, G., Watrigant, R.: A multivariate approach for checking resiliency in access control. In: Dondi, R., Fertin, G., Mauri, G. (eds.) AAIM 2016. LNCS, vol. 9778, pp. 173–184. Springer, Cham (2016). doi: 10.1007/978-3-319-41168-2_15 CrossRefGoogle Scholar
  6. 6.
    Crampton, J., Gutin, G., Watrigant, R.: Resiliency policies in access control revisited. In: Proceedings SACMAT 2016, pp. 101–111. ACM (2016)Google Scholar
  7. 7.
    Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Switzerland (2015)CrossRefMATHGoogle Scholar
  8. 8.
    Dorn, B., Schlotter, I.: Multivariate complexity analysis of swap bribery. Algorithmica 64(1), 126–151 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity: Texts in Computer Science. Springer, London (2013)CrossRefMATHGoogle Scholar
  10. 10.
    Eisenbrand, F., Shmonin, G.: Parametric integer programming in fixed dimension. Math. Oper. Res 33(4), 839–850 (2008)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Etscheid, M., Kratsch, S., Mnich, M., Röglin, H.: Polynomial kernels for weighted problems. J. Comput. Syst. Sci. 84, 1–10 (2017)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Frank, A., Tardos, É.: An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica 7(1), 49–65 (1987)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gramm, J., Niedermeier, R., Rossmanith, P.: Fixed-parameter algorithms for CLOSEST STRING and related problems. Algorithmica 37(1), 25–42 (2003)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Hemmecke, R., Onn, S., Romanchuk, L.: \(n\)-fold integer programming in cubic time. Math. Prog. 137(1), 325–341 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Jansen, B.M.P., Kratsch, S.: A structural approach to kernels for ILPs: treewidth and total unimodularity. In: Bansal, N., Finocchi, I. (eds.) ESA 2015. LNCS, vol. 9294, pp. 779–791. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-48350-3_65 CrossRefGoogle Scholar
  16. 16.
    Jansen, K., Kratsch, S., Marx, D., Schlotter, I.: Bin packing with fixed number of bins revisited. J. Comput. Syst. Sci. 79(1), 39–49 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kannan, R.: Minkowski’s convex body theorem and integer programming. Math. Oper. Res. 12(3), 415–440 (1987)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lenstra, H.W.: Integer programming with a fixed number of variables. Math. Op. Res. 8(4), 538–548 (1983)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Li, N., Tripunitara, M.V., Wang, Q.: Resiliency policies in access control. ACM Trans. Inf. Syst. Secur. 12(4), 113–137 (2009)CrossRefGoogle Scholar
  20. 20.
    Lokshtanov, D.: Parameterized integer quadratic programming: variables and coefficients. CoRR, abs/1511.00310 (2015)Google Scholar
  21. 21.
    Mnich, M., Wiese, A.: Scheduling and fixed-parameter tractability. Math. Program. 154(1), 533–562 (2014)MathSciNetMATHGoogle Scholar
  22. 22.
    Pevzner, P.: Computational Molecular Biology: An Algorithmic Approach. MIT Press, Cambridge (2000)MATHGoogle Scholar
  23. 23.
    Sebő, A.: Integer plane multiflows with a fixed number of demands. J. Comb. Theory Ser. B 59(2), 163–171 (1993)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Jason Crampton
    • 1
  • Gregory Gutin
    • 1
  • Martin Koutecký
    • 2
  • Rémi Watrigant
    • 3
  1. 1.Royal HollowayUniversity of LondonEghamUK
  2. 2.Department of Applied MathematicsCharles UniversityPragueCzech Republic
  3. 3.Inria Sophia Antipolis MéditerranéeSophia-Antipolis CedexFrance

Personalised recommendations