On Kernelization and Approximation for the Vector Connectivity Problem

In the Vector Connectivity problem we are given an undirected graph G= (V, E) , a demand function λ: V→ { 0 , … , d} , and an integer k. The question is whether there exists a set S of at most k vertices such that every vertex v∈ V\ S has at least λ(v) vertex-disjoint paths to S; this abstractly captures questions about placing servers or warehouses relative to demands. The problem is 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a log-factor approximation (Boros et al., Networks ’14), and is fixed-parameter tractable in terms of k (Lokshtanov, unpublished ’14).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 Vectord-Connectivity where the upper bound d on demands is a fixed constant. For Vectord-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 opt-approximation and we can show that it has no kernelization to size polynomial in k or even k+ d unless NP⊆ coNP/ poly, which shows that f(d) poly (k) is optimal for Vectord-Connectivity. Finally, we give a simple randomized fixed-parameter algorithm for Vector Connectivity with respect to k based on matroid intersection.