Abstract
Given an instance of an optimization problem and a good solution of that instance, the reoptimization is a concept of analyzing how does the solution change when the instance is locally modified. We investigate reoptimization of the following problems: Maximum Weighted Independent Set, Maximum Weighted Clique, Minimum Weighted Dominating Set, Minimum Weighted Set Cover and Minimum Weighted Vertex Cover. The local modifications we consider are addition or removal of a constant number of edges to the graph, or elements to the covering sets in case of Set Cover problem. We present the following results:
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We provide a PTAS for reoptimization of the unweighted versions of the aforementioned problems when the input solution is optimal.
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We provide two general techniques for analyzing approximation ratio of the weighted reoptimization problems.
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We apply our techniques to reoptimization of the considered optimization problems and obtain tight approximation ratios in all the cases.
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References
Archetti, C., Bertazzi, L., Speranza, M.G.: Reoptimizing the traveling salesman problem. Networks 42(3), 154–159 (2003)
Ausiello, G., Escoffier, B., Monnot, J., Paschos, V.T.: Reoptimization of minimum and maximum traveling salesman’s tours. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 196–207. Springer, Heidelberg (2006)
Bar-Yehuda, R., Even, S.: A local-ratio theorem for approximating the weighted vertex cover problem. Annals of Discrete Mathematics 25, 27–46 (1985)
Bilò, D., Böckenhauer, H.-J., Hromkovič, J., Královič, R., Mömke, T., Widmayer, P., Zych, A.: Reoptimization of steiner trees. In: Gudmundsson, J. (ed.) SWAT 2008. LNCS, vol. 5124, pp. 258–269. Springer, Heidelberg (2008)
Böckenhauer, H.-J., Forlizzi, L., Hromkovič, J., Kneis, J., Kupke, J., Proietti, G., Widmayer, P.: Reusing optimal TSP solutions for locally modified input instances (extended abstract). In: Fourth IFIP International Conference on Theoretical Computer Science—TCS 2006. IFIP Int. Fed. Inf. Process, vol. 209, pp. 251–270. Springer, New York (2006)
Böckenhauer, H.-J., Hromkovic, J., Mömke, T., Widmayer, P.: On the hardness of reoptimization. In: Geffert, V., Karhumäki, J., Bertoni, A., Preneel, B., Návrat, P., Bieliková, M. (eds.) SOFSEM 2008. LNCS, vol. 4910, pp. 50–65. Springer, Heidelberg (2008)
Boppana, R., Halldórsson, M.M.: Approximating maximum independent sets by excluding subgraphs. BIT 32(2), 180–196 (1992)
Escoffier, B., Milanic, M., Paschos, V.T.: Simple and fast reoptimizations for the Steiner tree problem. Technical Report 2007-01, DIMACS (2007)
Hastad, J.: Clique is hard to approximate within n 1 − ε. In: Foundations of Computer Science -FOCS 1996, pp. 627–636 (1996)
Hastad, J.: Some optimal inapproximability results. In: STOC 1997: Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, pp. 1–10. ACM, New York (1997)
Johnson, D.S.: Approximation algorithms for combinatorial problems. In: STOC 1973: Proceedings of the fifth annual ACM symposium on Theory of computing, pp. 38–49. ACM, New York (1973)
Monien, B., Speckenmeyer, E.: Ramsey numbers and an approximation algorithm for the vertex cover problem. Acta Inf. 22(1), 115–123 (1985)
Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability pcp characterization of np. In: STOC 1997: Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, pp. 475–484. ACM, New York (1997)
Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. In: STOC 2006: Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pp. 681–690. ACM, New York (2006)
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Bilò, D., Widmayer, P., Zych, A. (2009). Reoptimization of Weighted Graph and Covering Problems. In: Bampis, E., Skutella, M. (eds) Approximation and Online Algorithms. WAOA 2008. Lecture Notes in Computer Science, vol 5426. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-93980-1_16
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DOI: https://doi.org/10.1007/978-3-540-93980-1_16
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