On Kernelization and Approximation for the Vector Connectivity Problem

Stefan Kratsch, Manuel Sorge

Abstract: In the Vector Connectivity problem we are given an undirected graph \(G=(V,E)\), a demand function \(\lambda :V\rightarrow \{0,\ldots ,d\}\), and an integer k. The question is whether there exists a set S of at most k vertices such that every vertex \(v\in V{\setminus } S\) has at least \(\lambda (v)\) vertex-disjoint paths to S; this abstractly captures questions about placing servers or warehouses relative to demands. The problem is \(\mathsf {NP}\)-hard already for instances with \(d=4\) (Cicalese et al., Theoretical Computer Science ’15), admits a log-factor approximation (Boros et al., Networks ’14), and is fixed-parameter tractable in terms of k (Lokshtanov, unpublished ’14). We prove several results regarding kernelization and approximation for Vector Connectivity and the variant Vector d-Connectivity where the upper bound d on demands is a fixed constant. For Vector d-Connectivity we give a factor d-approximation algorithm and construct a vertex-linear kernelization, that is, an efficient reduction to an equivalent instance with \(f(d)k=O(k)\) vertices. For Vector Connectivity we have a factor \(\mathsf {opt} \)-approximation and we can show that it has no kernelization to size polynomial in k or even \(k+d\) unless \(\mathsf {NP} \subseteq \mathsf {coNP}/\mathsf {poly}\), which shows that \(f(d){\text {poly}}(k)\) is optimal for Vector d-Connectivity. Finally, we give a simple randomized fixed-parameter algorithm for Vector Connectivity with respect to k based on matroid intersection.