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- W2953301105 abstract "We study the following general disjoint paths problem: given a supply graph $G$, a set $Tsubseteq V(G)$ of terminals, a demand graph $H$ on the vertices $T$, and an integer $k$, the task is to find a set of $k$ pairwise vertex-disjoint valid paths, where we say that a path of the supply graph $G$ is valid if its endpoints are in $T$ and adjacent in the demand graph $H$. For a class $mathcal{H}$ of graphs, we denote by $mathcal{H}$-Maximum Disjoint Paths the restriction of this problem when the demand graph $H$ is assumed to be a member of $mathcal{H}$. We study the fixed-parameter tractability of this family of problems, parameterized by $k$. Our main result is a complete characterization of the fixed-parameter tractable cases of $mathcal{H}$-Maximum Disjoint Paths for every hereditary class $mathcal{H}$ of graphs: it turns out that complexity depends on the existence of large induced matchings and large induced skew bicliques in the demand graph $H$ (a skew biclique is a bipartite graph on vertices $a_1$, $dots$, $a_n$, $b_1$, $dots$, $b_n$ with $a_i$ and $b_j$ being adjacent if and only if $ile j$). Specifically, we prove the following classification for every hereditary class $mathcal{H}$. 1. If $mathcal{H}$ does not contain every matching and does not contain every skew biclique, then $mathcal{H}$-Maximum Disjoint Paths is FPT. 2. If $mathcal{H}$ does not contain every matching, but contains every skew biclique, then $mathcal{H}$-Maximum Disjoint Paths is W[1]-hard, admits an FPT approximation, and the valid paths satisfy an analog of the Erdős-Posa property. 3. If $mathcal{H}$ contains every matching, then $mathcal{H}$-Maximum Disjoint Paths is W[1]-hard and the valid paths do not satisfy the analog of the Erdős-Posa property." @default.
- W2953301105 created "2019-06-27" @default.
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- W2953301105 date "2014-12-22" @default.
- W2953301105 modified "2023-09-27" @default.
- W2953301105 title "An exact characterization of tractable demand patterns for maximum disjoint path problems" @default.
- W2953301105 doi "https://doi.org/10.1137/1.9781611973730.44" @default.
- W2953301105 hasPublicationYear "2014" @default.
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