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# Half-Integral Linkages in Highly Connected Directed Graphs

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LIPIcs.ESA.2017.36.pdf
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## Cite As

Katherine Edwards, Irene Muzi, and Paul Wollan. Half-Integral Linkages in Highly Connected Directed Graphs. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 36:1-36:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ESA.2017.36

## Abstract

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.
##### Keywords
• directed graph
• treewidth

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