Defensive Alliances in Graphs of Bounded Treewidth

TL;DR

This paper proves that finding minimum defensive alliances in graphs is W[1]-hard when parameterized by treewidth, showing it is unlikely to have fixed-parameter tractable algorithms despite the parameter's prominence.

Contribution

It establishes the W[1]-hardness of the defensive alliance problem parameterized by treewidth, resolving an open question and contrasting with known FPT results for related parameters.

Findings

The problem is W[1]-hard when parameterized by treewidth.

This rules out fixed-parameter tractable algorithms under common assumptions.

The proof uses techniques from hardness results for secure set problems.

Abstract

A set S of vertices of a graph is a defensive alliance if, for each element of S, the majority of its neighbors is in S. The problem of finding a defensive alliance of minimum size in a given graph is NP-hard and there are polynomial-time algorithms if certain parameters are bounded by a fixed constant. In particular, fixed-parameter tractability results have been obtained for some structural parameters such as the vertex cover number. However, for the parameter treewidth, the question of whether the problem is FPT has remained open. This is unfortunate because treewidth is perhaps the most prominent graph parameter and has proven successful for many problems. In this work, we give a negative answer by showing that the problem is W[1]-hard when parameterized by treewidth, which rules out FPT algorithms under common assumptions. This is surprising since the problem is known to be FPT when parameterized by solution size and "subset problems" that satisfy this property usually tend to be FPT for bounded treewidth as well. We prove W[1]-hardness by using techniques from a recent hardness result for the problem of finding so-called secure sets in a graph.