Abstract
A digraph G = (V,E) with a distinguished set T ⊆ V of terminals is called inner Eulerian if for each v ∈ V − T the numbers of arcs entering and leaving v are equal. By a T-path we mean a simple directed path connecting distinct terminals with all intermediate nodes in V − T. This paper concerns the problem of finding a maximum T-path packing, i.e. a maximum collection of arc-disjoint T-paths.
A min-max relation for this problem was established by Lomonosov. The capacitated version was studied by Ibaraki, Karzanov, and Nagamochi, who came up with a strongly-polynomial algorithm of complexity O(φ(V,E) ·logT + V 2 E) (hereinafter φ(n,m) denotes the complexity of a max-flow computation in a network with n nodes and m arcs).
For unit capacities, the latter algorithm takes O(φ(V,E) ·logT + VE) time, which is unsatisfactory since a max-flow can be found in o(VE) time. For this case, we present an improved method that runs in O(φ(V,E) ·logT + E logV) time. Thus plugging in the max-flow algorithm of Dinic, we reduce the overall complexity from O(VE) to O( min (V 2/3 E, E 3/2) ·logT).
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References
Babenko, M.A., Karzanov, A.V.: Free multiflows in bidirected and skew-symmetric graphs. Discrete Applied Mathematics 155, 1715–1730 (2007)
Cherkassky, B.V.: A solution of a problem on multicommodity flows in a network. Ekonomika i Matematicheskie Metody 13(1), 143–151 (1977) (in Russian)
Dinic, E.A.: Algorithm for solution of a problem of maximum flow in networks with power estimation. Dokl. Akad. Nauk. SSSR 194, 754–757 (in Russian) (translated in Soviet Math. Dokl. 111, 277–279)
Edmonds, J., Johnson, E.L.: Matching: a well-solved class of integer linear programs. In: Guy, R., Hanani, H., Sauer, N., Schönhein, J. (eds.) Combinatorial Structures and Their Applications, pp. 89–92. Gordon and Breach, NY (1970)
Even, S., Tarjan, R.E.: Network Flow and Testing Graph Connectivity. SIAM Journal on Computing 4, 507–518 (1975)
Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton Univ. Press, Princeton (1962)
Fortune, S., Hopcroft, J., Wyllie, J.: The directed subgraph homeomorphism problem. Theoretical Computer Sci. 10, 111–121 (1980)
Frank, A.: On connectivity properties of Eulerian digraphs. Ann. Discrete Math. 41, 179–194 (1989)
Goldberg, A.V., Karzanov, A.V.: Path problems in skew-symmetric graphs. Combinatorica 16, 129–174 (1996)
Goldberg, A.V., Karzanov, A.V.: Maximum skew-symmetric flows and matchings. Mathematical Programming 100(3), 537–568 (2004)
Goldberg, A.V., Rao, S.: Beyond the flow decomposition barrier. In: Proc. 38th IEEE Symposium Foundations of Computer Science (1997); adn Journal of the ACM 45, 783–797 (1998)
Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum flow problem. J. ACM 35, 921–940 (1988)
Ibaraki, T., Karzanov, A.V., Nagamochi, H.: A fast algorithm for finding a maximum free multiflow in an inner Eulerian network and some generalizations. Combinatorica 18(1), 61–83 (1998)
Karger, D.R.: Random sampling in cut, flow, and network design problems. Mathematics of Operations Research (1998)
Karger, D.R.: Better random sampling algorithms for flows in undirected graphs. In: Proc. 9th Annual ACM+SIAM Symposium on Discrete Algorithms, pp. 490–499 (1998)
Karger, D.R., Levine, M.S.: Finding Maximum flows in undirected graphs seems easier than bipartite matching. In: Proc. 30th Annual ACM Symposium on Theory of Computing, pp. 69–78 (1997)
Karzanov, A.V.: O nakhozhdenii maksimalnogo potoka v setyakh spetsialnogo vida i nekotorykh prilozheniyakh. In: Matematicheskie Voprosy Upravleniya Proizvodstvom, vol. 5. University Press (1973) (in Russian)
Karzanov, A.V.: Combinatorial methods to solve cut-dependent problems on multiflows. In: Combinatorial Methods for Flow Problems, Inst. for System Studies, Moscow, vol. (3), pp. 6–69 (1979) (in Russian)
Karzanov, A.V.: Fast algorithms for solving two known problems on undirected multicommodity flows. In: Combinatorial Methods for Flow Problems, Inst. for System Studies, Moscow, vol. (3), pp. 96–103 (1979) (in Russian)
Kupershtokh, V.L.: A generalization of Ford-Fulkerson theorem to multiterminal networks. Kibernetika 7(3), 87–93 (1971) (in Russian) (Translated in Cybernetics 7(3) 494-502)
Lawler, E.L.: Combinatorial Optimization: Networks and Matroids. Holt, Reinhart, and Winston, NY (1976)
Lovász, L.: On some connectivity properties of Eulerian graphs. Acta Math. Akad. Sci. Hung. 28, 129–138 (1976)
Lovász, L.: Matroid matching and some applications. J. Combinatorial Theory, Ser. B 28, 208–236 (1980)
Mader, W.: Über die Maximalzahl kantendisjunkter A-Wege. Archiv der Mathematik (Basel) 30, 325–336 (1978)
Schrijver, A.: Combinatorial Optimization. Springer (2003)
Sleator, D.D., Tarjan, R.E.: A data structure for dynamic trees. J. Comput. Syst. Sci. 26(3), 362–391 (1983)
Tutte, W.T.: Antisymmetrical digraphs. Canadian J. Math. 19, 1101–1117 (1967)
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Babenko, M.A., Salikhov, K., Artamonov, S. (2012). An Improved Algorithm for Packing T-Paths in Inner Eulerian Networks. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds) Computing and Combinatorics. COCOON 2012. Lecture Notes in Computer Science, vol 7434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32241-9_10
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