Abstract
In this paper, we study a variant of set packing, in which a set P of paths in a graph \(G=(V,E)\) is given, the goal is to find a maximum number of edge-disjoint paths of P. We show that the problem is NP-hard even if each path in P contains at most three edges, while it is hard to approximate within \(O(|E|^{1/2-\epsilon })\) for the general case unless \(NP=ZPP\). In the positive aspect, a parameterized algorithm relying on the maximum degree and the tree-width of G is derived. For tree networks, we present a polynomial time optimal algorithm.
This work was partially supported by NSFC Grant 11531014.
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References
Horing, S., Menard, J., Staehler, R., Yokelson, B.: Stored program controlled network: overview. Bell Syst. Tech. J. 61(7), 1579–1588 (1982)
Kuipers, F.A.: An overview of algorithms for network survivability. ISRN Commun. Netw. 2012, 24 (2012)
Michael, R.G., David, S.J.: Computers and Intractability: A Guide to the Theory of NP-Completeness, pp. 90–91. W. H. Freeman and Company, San Francisco (1979)
Håstad, J.: Clique is hard to approximate within \(n^{1-\epsilon }\). In: Acta Mathematica, pp. 627–636 (1996)
Halldórsson, M.Ú.M., KratochvÃl, J., Telle, J.A.: Independent sets with domination constraints. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 176–187. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0055051
Halldórsson, M.Ú.M.: Approximations of independent sets in graphs. In: Jansen, K., Rolim, J. (eds.) APPROX 1998. LNCS, vol. 1444, pp. 1–13. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0053959
Hurkens, C.A.J., Schrijver, A.: On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems. SIAM J. Discrete Math. 2(1), 68–72 (1989)
Halldórsson, M.Ú.M.: Approximating discrete collections via local improvements. In: Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 160–169 (1995)
Cygan, M., Grandoni, F., Mastrolilli, M.: How to sell hyperedges: the hypermatching assignment problem. In: Proceedings of the Twenty-Fourth Annual ACM-SIAM Aymposium on Discrete Algorithms, SIAM, pp. 342–351 (2013)
Sviridenko, M., Ward, J.: Large neighborhood local search for the maximum set packing problem. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013. LNCS, vol. 7965, pp. 792–803. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39206-1_67
Fellows, M.R., Downey, R.G.: Parameterized Complexity. Springer, New York (1999). https://doi.org/10.1007/978-1-4612-0515-9
Jia, W., Zhang, C., Chen, J.: An efficient parameterized algorithm for m-set packing. J. Algorithms 50(1), 106–117 (2004)
Koutis, I.: A faster parameterized algorithm for set packing. Inf. Process. Lett. 94(1), 7–9 (2005)
Fellows, M.R., Knauer, C., Nishimura, N., Ragde, P., Rosamond, F., Stege, U., Thilikos, D.M., Whitesides, S.: Faster fixed-parameter tractable algorithms for matching and packing problems. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 311–322. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-30140-0_29
Abu-Khzam, F.N.: An improved kernelization algorithm for r-set packing. Inf. Process. Lett. 110(16), 621–624 (2010)
Chen, L., Ye, D., Zhang, G.: An improved lower bound for rank four scheduling. Oper. Res. Lett. 42(5), 348–350 (2014)
Micali, S., Vazirani, V.V.: An O(\(\sqrt{|V|}|{E}|\)) algorithm for finding maximum matching in general graphs. In: Proceedings of the Twenty-First Annual Symposium on Foundations of Computer Science, pp. 17–27 (1980)
Bodlaender, H.L., Kloks, T.: Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms 21(2), 358–402 (1996)
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Xu, C., Zhang, G. (2018). The Path Set Packing Problem. In: Wang, L., Zhu, D. (eds) Computing and Combinatorics. COCOON 2018. Lecture Notes in Computer Science(), vol 10976. Springer, Cham. https://doi.org/10.1007/978-3-319-94776-1_26
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