Abstract
Given a tree T with costs on edges and a collection of terminal sets X = {S 1, S 2, ..., S l }, the generalized Multicut problem asks to find a set of edges on T whose removal cuts every terminal set in X, such that the total cost of the edges is minimized. The standard version of the problem can be approximately solved by reducing it to the classical Multicut on trees problem. For the prize-collecting version of the problem, we give a primal-dual 3-approximation algorithm and a randomized 2.55-approximation algorithm (the latter can be derandomized). For the k-version of the problem, we show an interesting relation between the problem and the Densest k-Subgraph problem, implying that approximating the k-version of the problem within O(n 1/6 − ε) for some small ε> 0 is hard. We also give a min { 2(l − k + 1), k }-approximation algorithm for the k-version of the problem via a nonuniform approach.
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References
Avidor, A., Langberg, M.: The multi-multiway cut problem. In: Proceedings of the 9th Scandinavian Workshop on Algorithm Theory (SWAT’04), pp. 273–284 (2004)
Feige, U.: Relations between average case complexity and approximation complexity. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC’02), pp. 534–543 (2002)
Garg, N., Vazirani, V., Yannakakis, M.: Approximate max-flow min-(multi)cut theorems and their applications. SIAM Journal on Computing 25, 235–251 (1996)
Garg, N., Vazirani, V., Yannakakis, M.: Primal-dual approximation algorithm for integral flow and multicut in trees. Algorithmica 18, 3–20 (1997)
Golovin, D., Nagarajan, V., Singh, M.: Approximating the k-multicut problem. In: Proceedings of the 17th ACM-SIAM Symposium on Discrete Algorithms (SODA’06), pp. 621–630 (2006)
Hajiaghayi, M., Jain, K.: The prize-collecting generalized Steiner tree problem via a new approach of primal-dual schema. In: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’06), pp. 631–640 (2006)
Jain, K., Mahdian, M., Markakis, E., Saberi, A., Vazirani, V.: Greedy facility location algorithms analyzied using dual fitting with factor-revealing LP. Journal of the ACM 50(6), 795–824 (2003)
Levin, A., Segev, D.: Partial multicuts in trees. In: Erlebach, T., Persinao, G. (eds.) WAOA 2005. LNCS, vol. 3879, pp. 320–333. Springer, Heidelberg (2006)
Khot, S.: Ruling out PTAS for graph min-bisection, densest subgraph and bipartite clique. In: Proceedings of the 44th Annual IEEE Symposium on the Foundations of Computer Science (FOCS’04), pp. 136–145 (2004)
Könemann, J., Parekh, O., Segev, D.: A unified approach to approximating partial covering problems. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 468–479. Springer, Heidelberg (2006)
Vazirani, V.: Approximation Algorithms, 2nd edn. Springer, Berlin (2003)
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Zhang, P. (2007). Approximating Generalized Multicut on Trees. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds) Computation and Logic in the Real World. CiE 2007. Lecture Notes in Computer Science, vol 4497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73001-9_85
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DOI: https://doi.org/10.1007/978-3-540-73001-9_85
Publisher Name: Springer, Berlin, Heidelberg
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