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- W4225536394 abstract "For the well-known Survivable Network Design Problem (SNDP) we are given an undirected graph $G$ with edge costs, a set $R$ of terminal vertices, and an integer demand $d_{s,t}$ for every terminal pair $s,tin R$. The task is to compute a subgraph $H$ of $G$ of minimum cost, such that there are at least $d_{s,t}$ disjoint paths between $s$ and $t$ in $H$. If the paths are required to be edge-disjoint we obtain the edge-connectivity variant (EC-SNDP), while internally vertex-disjoint paths result in the vertex-connectivity variant (VC-SNDP). Another important case is the element-connectivity variant (LC-SNDP), where the paths are disjoint on edges and non-terminals. In this work we shed light on the parameterized complexity of the above problems. We consider several natural parameters, which include the solution size $ell$, the sum of demands $D$, the number of terminals $k$, and the maximum demand $d_max$. Using simple, elegant arguments, we prove the following results. - We give a complete picture of the parameterized tractability of the three variants w.r.t. parameter $ell$: both EC-SNDP and LC-SNDP are FPT, while VC-SNDP is W[1]-hard. - We identify some special cases of VC-SNDP that are FPT: * when $d_maxleq 3$ for parameter $ell$, * on locally bounded treewidth graphs (e.g., planar graphs) for parameter $ell$, and * on graphs of treewidth $tw$ for parameter $tw+D$. - The well-known Directed Steiner Tree (DST) problem can be seen as single-source EC-SNDP with $d_max=1$ on directed graphs, and is FPT parameterized by $k$ [Dreyfus & Wagner 1971]. We show that in contrast, the 2-DST problem, where $d_max=2$, is W[1]-hard, even when parameterized by $ell$." @default.
- W4225536394 created "2022-05-05" @default.
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- W4225536394 date "2021-11-03" @default.
- W4225536394 modified "2023-09-23" @default.
- W4225536394 title "The Parameterized Complexity of the Survivable Network Design Problem" @default.
- W4225536394 doi "https://doi.org/10.48550/arxiv.2111.02295" @default.
- W4225536394 hasPublicationYear "2021" @default.
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