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- W4300696541 abstract "We study the half-integral $k$-Directed Disjoint Paths Problem ($tfrac12$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 $Lgeq 1$. 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 $kgeq 2$ there exists $L(k)$ such that $tfrac12$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 $tfrac12$kDDPP is solvable in time $O(|V(G)|^{f(k)})$ in $(36k^3+2k)$-strongly connected directed graphs. We also 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 $epsilon k$." @default.
- W4300696541 created "2022-10-04" @default.
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- W4300696541 date "2016-11-03" @default.
- W4300696541 modified "2023-09-28" @default.
- W4300696541 title "Half-integral linkages in highly connected directed graphs" @default.
- W4300696541 doi "https://doi.org/10.48550/arxiv.1611.01004" @default.
- W4300696541 hasPublicationYear "2016" @default.
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