Abstract
In this paper, we study the approximability of the capacitated network design problem (Cap-NDP) on undirected graphs: Given G = (V,E) with non-negative costs c and capacities u on its edges, source-sink pairs (s i , t i ) with demand r i , the goal is to find the minimum cost subgraph where the minimum (s i , t i ) cut with u-capacities is at least r i . When u ≡ 1, we get the usual SNDP for which Jain gave a 2-approximation algorithm [9]. Prior to our work, the approximability of undirected Cap-NDP was not well understood even in the single source-sink pair case. In this paper, we show that the single-source pair Cap-NDP is label-cover hard in undirected graphs.
An important special case of single source-sink pair undirected Cap-NDP is the following source location problem. Given an undirected graph, a collection of sources S and a sink t, find the minimum cardinality subset S′ ⊆ S such that flow(S′,t), the maximum flow from S′ to t, equals flow(S,t). In general, the problem is known to be set-cover hard. We give a O(ρ)-approximation when flow(s,t) ≈ ρ flow(s′,t) for s, s′ ∈ S, that is, all sources have max-flow values to the sink within a multiplicative ρ factor of each other.
The main technical ingredient of our algorithmic result is the following theorem which may have other applications. Given a capacitated, undirected graph G with a dedicated sink t, call a subset X ⊆ V irreducible if the maximum flow f(X) from X to t is strictly greater than that from any strict subset X′ ⊂ X, to t. We prove that for any irreducible set, X, the flow \(f(X)\geq\frac{1}{2}\sum_{i\in X} f_i\), where f i is the max-flow from i to t. That is, undirected flows are quasi-additive on irreducible sets.
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References
Andreev, K., Garrod, C., Golovin, D., Maggs, B., Meyerson, A.: Simultaneous source location. Trans. on Algorithms (TALG) 6(1), 1–17 (2009)
Arata, K., Iwata, S., Makino, K., Fujishige, S.: Locating sources to meet flow demands in undirected networks. J. of Algorithms 42, 54–68 (2002)
Chakrabarty, D., Chekuri, C., Khanna, S., Korula, N.: Approximability of capacitated network design. In: Günlük, O., Woeginger, G.J. (eds.) IPCO 2011. LNCS, vol. 6655, pp. 78–91. Springer, Heidelberg (2011)
Dodis, Y., Khanna, S.: Design networks with bounded pairwise distance. In: ACM Symp. on Theory of Computing, STOC (1999)
Even, G., Kortsarz, G., Slany, W.: On Network Design: Fixed charge flows and the covering Steiner problem. Trans. on Algorithms (TALG) 1(1), 74–101 (2005)
Feige, U.: Vertex cover is hardest to approximate on regular graphs, Tech. report, Weizmann Institute (2004)
Hagerup, T., Katajainen, J., Nishimura, N., Ragde, P.: Characterizations of multiterminal flow networks and computing flows in networks of bounded treewidth. Journal of Computer and System Sciences (JCSS) 57, 366–375 (1998)
Hajiaghayi, M.T., Khandekar, R., Kortsarz, G., Nutov, Z.: Capacitated Network Design problems: Hardness, approximation algorithms, and connections to group Steiner tree (2011), http://arxiv.org/pdf/1108.1176.pdf
Jain, K.: A factor 2 approximation algorithm for the generalized Steiner network problem. Combinatorica 21(1), 39–60 (2001)
Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2 − ε. In: Proceedings of FOCS (2003)
Kortsarz, G., Nutov, Z.: A note on two source location problems. J. of Discrete Algorithms 6(3), 520–525 (2008)
Labbe, M., Peeters, D., Thisse, J.-F.: Location on networks. In: Handbook in OR and MS, vol. 8, pp. 551–624 (1995)
Sakashita, M., Makino, K., Fujishige, S.: Minimum cost source location problems with flow requirements. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 769–780. Springer, Heidelberg (2006)
Tamura, H., Sengoku, M., Shinoda, S., Abe, T.: Location problems on undirected flow networks. IEICE Trans. E(73), 1989–1993 (1990)
Tamura, H., Sengoku, M., Shinoda, S., Abe, T.: Some covering problems in location theory on flow networks. IEICE Trans. E(75), 678–683 (1992)
Wolsey, L.: An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica 2(4), 385–393 (1982)
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Chakrabarty, D., Krishnaswamy, R., Li, S., Narayanan, S. (2013). Capacitated Network Design on Undirected Graphs. In: Raghavendra, P., Raskhodnikova, S., Jansen, K., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2013 2013. Lecture Notes in Computer Science, vol 8096. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40328-6_6
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DOI: https://doi.org/10.1007/978-3-642-40328-6_6
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