Abstract
An independence system \(\mathcal{F}\) is one of the most fundamental combinatorial concepts, which includes a variety of objects in graphs and hypergraphs such as matchings, stable sets, and matroids. We discuss the robustness for independence systems, which is a natural generalization of the greedy property of matroids. For a real number α > 0, a set \(X\in\mathcal{F}\) is said to be α-robust if for any k, it includes an α-approximation of the maximum k-independent set, where a set Y in \(\mathcal{F}\) is called k-independent if the size |Y| is at most k. In this paper, we show that every independence system has a \(1/\sqrt{\mu(\mathcal{F})}\)-robust independent set, where \(\mu(\mathcal{F})\) denotes the exchangeability of \(\mathcal{F}\). Our result contains a classical result for matroids and the ones of Hassin and Rubinstein,[12] for matchings and Fujita, Kobayashi, and Makino,[7] for matroid 2-intersections, and provides better bounds for the robustness for many independence systems such as b-matchings, hypergraph matchings, matroid p-intersections, and unions of vertex disjoint paths. Furthermore, we provide bounds of the robustness for nonlinear weight functions such as submodular and convex quadratic functions. We also extend our results to independence systems in the integral lattice with separable concave weight functions.
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References
Calinescu, G., Chekuri, C., Pal, M., Vondrak, J.: Maximizing a submodular set function subject to a matroid constraint. SIAM Journal on Computing (to appear)
Chan, Y.H., Lau, L.C.: On linear and semidefinite programming relaxations for hypergraph matching. In: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2010), pp. 1500–1511 (2010)
Chandra, B., Halldórsson, M.: Greedy local improvement and weighted set packing approximation. Journal of Algorithms 39, 223–240 (2001)
Edmonds, J.: Matroids and the greedy algorithm. Mathematical Programming 1, 127–136 (1971)
Fisher, M.L., Nemhauser, G.L., Wolsey, L.A.: An analysis of approximations for maximizing submodular set functions ii. Mathematical Programming Study 8, 73–87 (1978)
Frank, A.: How to make a digraph strongly connected. Combinatorica 1, 145–153 (1981)
Fujita, R., Kobayashi, Y., Makino, K.: Robust matchings and matroid intersections. In: de Berg, M., Meyer, U. (eds.) ESA 2010. LNCS, vol. 6347, pp. 123–134. Springer, Heidelberg (2010)
Fukunaga, T., Halldórsson, M., Nagamochi, H.: Robust cost colorings. In: Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2008), pp. 1204–1212 (2008)
Gabow, H.N.: An efficient reduction technique for degree-constrained subgraph and bidirected network flow problems. In: Proceedings of the 15th ACM Symposium on Theory of Computing (STOC 1983), pp. 448–456 (1983)
Goemans, M.X., Harvey, N.J.A., Iwata, S., Mirrokni, V.: Approximating submodular functions everywhere. In: Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2009), pp. 535–544 (2009)
Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, 2nd edn. Springer, Heidelberg (1993)
Hassin, R., Rubinstein, S.: Robust matchings. SIAM Journal on Discrete Mathematics 15, 530–537 (2002)
Hassin, R., Segev, D.: Robust subgraphs for trees and paths. ACM Transaction on Algorithms 2, 263–281 (2006)
Hásted, J.: Clique is hard to approximate within n 1 − ε. Acta Mathematica 182, 105–142 (1999)
Jenkyns, T.A.: The efficacy of the “greedy” algorithm. In: Proceedings of the 7th Southeastern Conference on Combinatorics, Graph Theory and Computing, pp. 341–350 (1976)
Jenkyns, T.A.: The greedy traveling salesman’s problem. Networks 9, 363–373 (1979)
Jungnickel, D.: Graphs, Networks, and Algorithms, 2nd edn. Algorithms and Computation in Mathematics, vol. 5. Springer, Heidelberg (2002)
Kakimura, N., Makino, K.: Robust independence systems, Mathematical Engineering Technical Reports METR 2011-14, University of Tokyo (2011)
Korte, B., Hausmann, D.: An analysis of the greedy heuristic for independence systems. Annals of Discrete Mathematics 2, 65–74 (1978)
Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms. Springer, Heidelberg (2006)
Lin, G., Nagarajan, C., Rajarama, R., Williamson, D.: A general approach for incremental approximation and hierarchical clustering. SIAM Journal on Computing 39, 3633–3669 (2010)
Lucchesi, C.L.: A Minimax Equality for Directed Graphs, PhD thesis, University of Waterloo (1976)
Mestre, J.: Greedy in approximation algorithms. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 528–539. Springer, Heidelberg (2006)
Mettu, R.R., Plaxton, C.G.: The online median problem. SIAM Journal on Computing 32, 816–832 (2003)
Rado, R.: Note on independence relations. Proceedings of the London Mathematical Society 7, 300–320 (1957)
Schrijver, A.: Combinatorial Optimization — Polyhedra and Efficiency. Springer, Heidelberg (2003)
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Kakimura, N., Makino, K. (2011). Robust Independence Systems. In: Aceto, L., Henzinger, M., Sgall, J. (eds) Automata, Languages and Programming. ICALP 2011. Lecture Notes in Computer Science, vol 6755. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22006-7_31
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DOI: https://doi.org/10.1007/978-3-642-22006-7_31
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