The Parameterized Complexity of the Survivable Network Design Problem
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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,t∈ 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 ℓ, 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 ℓ: 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_max≤ 3 for parameter ℓ, * on locally bounded treewidth graphs (e.g., planar graphs) for parameter ℓ, 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 ℓ.
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