The Parameterized Complexity of the Survivable Network Design Problem

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 ds,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 ds,t disjoint paths between s and t in H. Depending on the type of disjointness we obtain several variants of SNDP that have been widely studied in the literature: 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, i.e., they may only share 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 , the sum of demands D, the number of terminals k, and the maximum demand dmax. 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 : both EC-SNDP and LC-SNDP are FPT, while VC-SNDP is W[1]-hard (even in the uniform single-source case with k = 3). • We identify some special cases of VC-SNDP that are FPT:when dmax ≤ 3 for parameter , on locally bounded treewidth graphs (e.g., planar graphs) for parameter , and on graphs of treewidth tw for parameter tw + D, which is in contrast to a result by Bateni et al. [JACM 2011] who show NP-hardness for tw = 3 (even if dmax = 1, i.e., the Steiner Forest problem). • The well-known Directed Steiner Tree (DST) problem can be seen as single-source EC-SNDP with dmax = 1 on directed graphs, and is FPT parameterized by k [Dreyfus & Wagner 1971]. We show that in contrast, the 2-DST problem, where dmax = 2, is W[1]-hard, even when parameterized by (which is always larger than k). * Supported by the Czech Science Foundation GA ČR (grant #19-27871X), and by the Center for Foundations of Modern Computer Science (Charles Univ. project UNCE/SCI/004).