Parameterized Resiliency Problems via Integer Linear Programming
Abstract
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.
Keywords
Integer Linear Program Close String Input String Prefer Candidate Authorization PolicyReferences
- 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.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.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)MathSciNetCrossRefzbMATHGoogle Scholar
- 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.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.Crampton, J., Gutin, G., Watrigant, R.: Resiliency policies in access control revisited. In: Proceedings SACMAT 2016, pp. 101–111. ACM (2016)Google Scholar
- 7.Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Switzerland (2015)CrossRefzbMATHGoogle Scholar
- 8.Dorn, B., Schlotter, I.: Multivariate complexity analysis of swap bribery. Algorithmica 64(1), 126–151 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
- 9.Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity: Texts in Computer Science. Springer, London (2013)CrossRefzbMATHGoogle Scholar
- 10.Eisenbrand, F., Shmonin, G.: Parametric integer programming in fixed dimension. Math. Oper. Res 33(4), 839–850 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
- 11.Etscheid, M., Kratsch, S., Mnich, M., Röglin, H.: Polynomial kernels for weighted problems. J. Comput. Syst. Sci. 84, 1–10 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
- 12.Frank, A., Tardos, É.: An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica 7(1), 49–65 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
- 13.Gramm, J., Niedermeier, R., Rossmanith, P.: Fixed-parameter algorithms for CLOSEST STRING and related problems. Algorithmica 37(1), 25–42 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
- 14.Hemmecke, R., Onn, S., Romanchuk, L.: \(n\)-fold integer programming in cubic time. Math. Prog. 137(1), 325–341 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
- 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.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)MathSciNetCrossRefzbMATHGoogle Scholar
- 17.Kannan, R.: Minkowski’s convex body theorem and integer programming. Math. Oper. Res. 12(3), 415–440 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
- 18.Lenstra, H.W.: Integer programming with a fixed number of variables. Math. Op. Res. 8(4), 538–548 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
- 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.Lokshtanov, D.: Parameterized integer quadratic programming: variables and coefficients. CoRR, abs/1511.00310 (2015)Google Scholar
- 21.Mnich, M., Wiese, A.: Scheduling and fixed-parameter tractability. Math. Program. 154(1), 533–562 (2014)MathSciNetzbMATHGoogle Scholar
- 22.Pevzner, P.: Computational Molecular Biology: An Algorithmic Approach. MIT Press, Cambridge (2000)zbMATHGoogle Scholar
- 23.Sebő, A.: Integer plane multiflows with a fixed number of demands. J. Comb. Theory Ser. B 59(2), 163–171 (1993)MathSciNetCrossRefzbMATHGoogle Scholar