The Parameterized Complexity of the Survivable Network Design Problem

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 \(s, t \in R\). 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 \(H\). 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 \(l\), 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 \(\mathsf{FPT}\), while VC-SNDP is \(\mathsf{W[1]}\)-hard (even in the single-source case with \(k = 3\)).
• We identify some special cases of VC-SNDP that are \(\mathsf{FPT}\):
  • when \(d_\max \leq 3\) for parameter \(l\),
  • on locally bounded treewidth graphs (e.g., planar graphs) for parameter \(l\), and
  • on graphs of treewidth \(\operatorname{tw}\) for parameter \(\operatorname{tw} + D\), which is in contrast to a result by Bateni et al. [JACM 2011] who show \(\mathsf{NP}\)-hardness for \(\operatorname{tw} = 3\) (even if \(d_\max = 1\), i.e., the Steiner Forest problem).
• 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 \(\mathsf{FPT}\) parameterized by \(k\) [Dreyfus & Wagner 1971]. We show that in contrast, the 2-DST problem, where \(d_\max = 2\), is \(\mathsf{W[1]}\)-hard, even when parameterized by \(l\) (which is always larger than \(k\)).

This is joint work with Anish Mukherjee and Erik Jan van Leeuwen.

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