Systematic Exploration of Larger Local Search Neighborhoods for the Minimum Vertex Cover Problem

Maximilian Katzmann, Christian Komusiewicz

Abstract: We investigate the potential of exhaustively exploring larger neighborhoods in local search algorithms for Minimum Vertex Cover. More precisely, we study whether, for moderate values of k , it is feasible and worthwhile to determine, given a graph G with vertex cover C , if there is a k -swap S such that ( C ∖ S ) ∪ ( S ∖ C ) is a smaller vertex cover of G . First, we describe an algorithm running in ∆ O(k) ⋅ n time for searching the k -swap neighborhood on n -vertex graphs with maximum degree ∆. Then, we demonstrate that, by devising additional pruning rules that decrease the size of the search space, this algorithm can be implemented so that it solves the problem quickly for k ≈ 20. Finally, we show that it is worthwhile to consider moderately-sized k -swap neighborhoods. For our benchmark data set, we show that when combining our algorithm with a hill-climbing approach, the solution quality improves quickly with the radius k of the local search neighborhood and that in most cases optimal solutions can be found by setting k