Dominating an s-t-Cut in a Network

Abstract

We study an optimization problem with applications in design and analysis of resilient communication networks: given two vertices s, t in a graph G = (V,E), find a vertex set X ⊂ V of minimum cardinality, such that X and its neighborhood constitute an s-t vertex separator. Although the problem naturally combines notions of graph connectivity and domination, its computational properties significantly differ from these relatives.

In particular, we show that on general graphs the problem cannot be approximated to within a factor of $2^{\log^{1-\delta}{n}}$, with δ = 1 / loglog^c n and arbitrary $c<\frac{1}{2}$ (if P ≠ NP). This inapproximability result even applies if the subgraph induced by a solution set has the additional constraint of being connected. Furthermore, we give a $2\sqrt{n}$-approximation algorithm and study the problem on graphs with bounded node degree. With Δ being the maximum degree of nodes V ∖ {s,t}, we identify a (Δ + 1) approximation algorithm.