Effective and Efficient Data Reduction for the Subset Interconnection Design Problem

Abstract

The NP-hard Subset Interconnection Design problem is motivated by applications in designing vacuum systems and scalable overlay networks. It has as input a set V and a collection of subsets V ₁, V ₂, …, V _m , and asks for a minimum-cardinality edge set E such that for the graph G = (V,E) all induced subgraphs G[V ₁], G[V ₂], …, G[V _m ] are connected. It has also been studied under the name Minimum Topic-Connected Overlay. We study Subset Interconnection Design in the context of polynomial-time data reduction rules that preserve optimality. Our contribution is threefold: First, we point out flaws in earlier polynomial-time data reduction rules. Second, we provide a fixed-parameter tractability result for small subset sizes and tree-like output graphs. Third, we show linear-time solvability in case of a constant number m of subsets, implying fixed-parameter tractability for the parameter m. To achieve our results, we elaborate on polynomial-time data reduction rules (partly “repairing” previous flawed ones) which also may be of practical use in solving Subset Interconnection Design.

JC was supported by Studienstiftung des Deutschen Volkes, MS and MW were supported by Deutsche Forschungsgemeinschaft (projects NI 369/12 and NI 369/9), and part of the work of OS and MW was done while they were affiliated with TU Berlin.