Half-Integral Linkages in Highly Connected Directed Graphs
We study the half-integral k-Directed Disjoint Paths Problem (1/2 kDDPP) in highly strongly connected digraphs. The integral kDDPP is NP-complete even when restricted to instances where k=2, and the input graph is L-strongly connected, for any L >= 1. We show that when the integrality condition is relaxed to allow each vertex to be used in two paths, the problem becomes efficiently solvable in highly connected digraphs (even with k as part of the input).
Specifically, we show that there is an absolute constant c such that for each k >= 2 there exists L(k) such that 1/2 kDDPP is solvable in time O(|V(G)|^c) for a L(k)-strongly connected directed graph G. As the function L(k) grows rather quickly, we also show that 1/2 kDDPP is solvable in time O(|V(G)|^{f(k)}) in (36k^3+2k)-strongly connected directed graphs. We show that for each epsilon<1, deciding half-integral feasibility of kDDPP instances is NP-complete when k is given as part of the input, even when restricted to graphs with strong connectivity epsilon k.
linkage
directed graph
treewidth
36:1-36:12
Regular Paper
Katherine
Edwards
Katherine Edwards
Irene
Muzi
Irene Muzi
Paul
Wollan
Paul Wollan
10.4230/LIPIcs.ESA.2017.36
Marek Cygan, Daniel Marx, Marcin Pilipczuk, and Michal Pilipczuk. The planar directed k-vertex-disjoint paths problem is fixed-parameter tractable. In Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pages 197-206, Oct 2013. URL: http://dx.doi.org/10.1109/FOCS.2013.29.
http://dx.doi.org/10.1109/FOCS.2013.29
Reinhard Diestel, Tommy R Jensen, Konstantin Yu Gorbunov, and Carsten Thomassen. Highly connected sets and the excluded grid theorem. Journal of Combinatorial Theory, Series B, 75(1):61-73, 1999.
Katherine Edwards, Irene Muzi, and Paul Wollan. Half-integral linkages in highly connected directed graphs, 2016. URL: https://arxiv.org/abs/1611.01004.
https://arxiv.org/abs/1611.01004
Steven Fortune, John Hopcroft, and James Wyllie. The directed subgraph homeomorphism problem. Theoretical Computer Science, 10(2):111-121, 1980.
Thor Johnson, Neil Robertson, Paul D Seymour, and Robin Thomas. Directed tree-width. Journal of Combinatorial Theory, Series B, 82(1):138-154, 2001.
Ken-ichi Kawarabayashi. An improved algorithm for finding cycles through elements. In International Conference on Integer Programming and Combinatorial Optimization, pages 374-384. Springer, 2008.
Ken-ichi Kawarabayashi, Yusuke Kobayashi, and Stephan Kreutzer. An excluded half-integral grid theorem for digraphs and the directed disjoint paths problem. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, STOC'14, pages 70-78, New York, NY, USA, 2014. ACM.
Ken-ichi Kawarabayashi and Stephan Kreutzer. An excluded grid theorem for digraphs with forbidden minors. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 72-81. URL: http://dx.doi.org/10.1137/1.9781611973402.6.
http://dx.doi.org/10.1137/1.9781611973402.6
Stephan Kreutzer Ken-Ichi Kawarabayashi. The directed grid theorem, 2014. URL: https://arxiv.org/abs/1411.5681.
https://arxiv.org/abs/1411.5681
Stephan Kreutzer Ken-ichi Kawarabayashi. The directed grid theorem. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, pages 655-664. ACM, 2015.
Karl Menger. Zur allgemeinen Kurventheorie. Fundamenta Mathematicae, 1(10):96-115, 1927.
Bruce Reed. Introducing directed tree width. Electronic Notes in Discrete Mathematics, (3):1-8, 2000.
Neil Robertson and Paul D Seymour. Graph minors XIII. The disjoint paths problem. Journal of Combinatorial Theory, Series B, 63(1):65-110, 1995.
Alexander Schrijver. Finding k disjoint paths in a directed planar graph. SIAM Journal on Computing, 23(4):780-788, 1994.
Aleksandrs Slivkins. Parameterized tractability of edge-disjoint paths on directed acyclic graphs. SIAM Journal on Discrete Mathematics, 24(1):146-157, 2010.
Carsten Thomassen. 2-linked graphs. European Journal of Combinatorics, 1(4):371-378, 1980.
Carsten Thomassen. Highly connected non-2-linked digraphs. Combinatorica, 11(4):393-395, 1991.
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