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 φ: V → {0,., d}, and an integer κ. The question is whether there exists a set S of at most κ vertices such that every vertex ν ∈ V \ S has at least φ(ν) vertex-disjoint paths to S; this abstractly captures questions about placing servers in a network, or warehouses on a map, relative to demands. The problem is NP-hard already for instances with d = 4 (Cicalese et al., Theor. Comput. Sci. '15), admits a log-factor approximation (Boros et al., Networks '14), and is fixed-parameter tractable in terms of κ (Lokshtanov, unpublished '14).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 constant. For vector d-connectivity we give a factor d-approximation algorithm and construct a vertexlinear kernelization, i.e., an efficient reduction to an equivalent instance with f(d) κ = O(κ) vertices. For vector connectivity we get a factor opt-approximation and we show that it has no kernelization to size polynomial in κ+d unless NP⊆ coNP/poly, making f(d) poly(κ) optimal for vector d-connectivity. Finally, we provide a write-up for fixed-parameter tractability of vector connectivity (κ) by giving a different algorithm based on matroid intersection.