Polynomial-time data reduction for the subset interconnection design problem

The NP-hard SUBSET INTERCONNECTION DESIGN problem, also known as MINIMUM TOPIC-CONNECTED OVERLAY, is motivated by numerous applications including the design of scalable overlay networks and vacuum systems. It has as input a finite set V and a collection of subsets V₁, V₂, . . ., V_m ⊆ V, and asks for a minimum-cardinality edge set E such that for the graph G = (V, E) all induced subgraphs G[V₁], G[V₂], . . ., G[V_m] are connected. We study SUBSET INTERCONNECTION DESIGN in the context of polynomial-time data reduction rules that preserve the possibility of constructing optimal solutions. Our contribution is threefold: First, we show the incorrectness of earlier polynomial-time data reduction rules. Second, we show linear-time solvability in case of a constant number m of subsets, implying fixed-parameter tractability for the parameter m. Third, we provide a fixed-parameter tractability result for small subset sizes and tree-like output graphs. To achieve our results, we elaborate on polynomial-time data reduction rules which also may be of practical use in solving Subset Interconnection Design.