Abstract
Consider a suboptimal solution S for a maximization problem. Suppose S’s value is small compared to an optimal solution OPT to the problem, yet S is structurally similar to OPT. A natural question in this setting is whether there is a way of improving S based solely on this information. In this paper we introduce the Structural Continuous Greedy Algorithm, answering this question affirmatively in the setting of the Nonmonotone Submodular Maximization Problem. We improve on the best approximation factor known for this problem. In the Nonmonotone Submodular Maximization Problem we are given a non-negative submodular function f, and the objective is to find a subset maximizing f. Our method yields an 0.42-approximation for this problem, improving on the current best approximation factor of 0.41 given by Gharan and Vondrák [5]. On the other hand, Feige et al. [4] showed a lower bound of 0.5 for this problem.
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References
Calinescu, G., Chekuri, C., Pál, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. To appear in SIAM J. Comput.
Chekuri, C., Vondrák, J., Zenklusen, R.: Dependent randomized rounding via exchange properties of combinatorial structures. In: 51st Annual Symposium on Foundations of Computer Science, pp. 575–584. IEEE Computer Society, Washington DC (2010)
Chekuri, C., Vondrák, J., Zenklusen, R.: Submodular function maximization via the multilinear relaxation and contention resolution schemes. To appear in the 42nd ACM Symposium on Theory of Computer Science (2011)
Feige, U., Mirrokni, V.S., Vondrák, J.: Maximizing non-monotone submodular functions. In: 48th Annual IEEE Symposium on Foundations of Computer Science, pp. 461–471. IEEE Computer Society, Washington DC (2007)
Gharan, S.O., Vondrák, J.: Submodular maximization by simulated annealing. In: 22nd ACM-SIAM Symposium on Discrete Algorithms, pp. 1096–1116 (2011)
Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42(6), 1115–1145 (1995)
Gupta, A., Roth, A., Schoenebeck, G., Talwar, K.: Constrained non-monotone submodular maximization: Offline and secretary algorithms. In: Saberi, A. (ed.) WINE 2010. LNCS, vol. 6484, pp. 246–257. Springer, Heidelberg (2010)
Iwata, S., Nagano, K.: Submodular function minimization under covering constraints. In: 50th Annual IEEE Symposium on Foundations of Computer Science, pp. 671–680. IEEE Computer Society, Washington DC (2009)
Jegelka, S., Bilmes, J.: Cooperative cuts: Graph cuts with submodular edge weights. Technical report, Max Planck Institute for Biological Cybernetics (2010)
Kulik, A., Shachnai, H., Tamir, T.: Maximizing submodular set functions subject to multiple linear constraints. In: 20th ACM-SIAM Symposium on Discrete Algorithms, pp. 545–554. Society for Industrial and Applied Mathematics, Philadelphia (2009)
Lee, J., Mirrokni, V.S., Nagarajan, V., Sviridenko, M.: Maximizing non-monotone submodular functions under matroid or knapsack constraints. SIAM J. Discrete Mathematics 23(4), 2053–2078 (2010)
Lee, J., Sviridenko, M., Vondrák, J.: Submodular maximization over multiple matroids via generalized exchange properties. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) APPROX 2009. LNCS, vol. 5687, pp. 244–257. Springer, Heidelberg (2009)
Lovász, L.: Submodular functions and convexity. In: Bachem, A., Grötschel, M., Korte, B. (eds.) Mathematical Programming: the State of the Art, pp. 235–257. Springer, Berlin (1983)
Lovász, L., Grötschel, M., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatoria 1(2), 169–197 (1981)
Svitkina, Z., Fleischer, L.: Submodular approximation: Sampling-based algorithms and lower bounds. In: 49th Annual IEEE Symposium on Foundations of Computer Science, pp. 697–706. IEEE Computer Society, Washington DC (2008)
Vondrák, J.: Optimal approximation for the submodular welfare problem in the value oracle model. In: 40th ACM Symposium on Theory of Computer Science, pp. 67–74. ACM, New York (2008)
Vondrák, J.: Symmetry and approximability of submodular maximization problems. In: 50th Annual IEEE Symposium on Foundations of Computer Science, pp. 651–670. IEEE Computer Society, Washington DC (2009)
Wolsey, L.A.: An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica 2(4), 385–393 (1982)
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Feldman, M., Naor, J.(., Schwartz, R. (2011). Nonmonotone Submodular Maximization via a Structural Continuous Greedy Algorithm. 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_29
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DOI: https://doi.org/10.1007/978-3-642-22006-7_29
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