Finding 3-Swap-Optimal Independent Sets and Dominating Sets is Hard

Local search problems consist of a set of feasible solutions, an objective function, and a neighborhood relation which assigns to each feasible solution a set of neighbors. A solution S is called locally optimal if none of its neighbors has a better objective value than S . For PLS-complete local search problems, there is presumably no polynomial-time algorithm which finds a locally optimal solution, even though determining whether a solution is locally optimal and finding a better one if this is not the case can be done in polynomial time. We study local search for Weighted Independent Set and Weighted Dominating Set with the 3-swap neighborhood. The 3-swap neighborhood of a vertex set S in G is the collection of vertex sets which can be obtained from S by exchanging at most three vertices. On the negative side, the problem of finding a 3-swap-optimal independent set or dominating set is PLS-complete. Using the PLS-completeness for independent sets with the 3-swap-neighborhood, we also show PLS-completeness for a large class of maximum-weight subgraph problems. On the positive side, we show that locally optimal independent sets or dominating sets can be found in polynomial time when allowing all 3-swaps except (a) the swaps that remove two vertices from the current solution and add one vertex to the solution or (b) the swaps that remove one vertex from the current solution and add two vertices to the solution. This result is shown via general algorithms for subset-weight optimization problems with a restricted k -swap neighborhood.