Finding 3-Swap-Optimal Independent Sets and Dominating Sets Is Hard
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 replacing it by 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 set of vertex sets which can be obtained from S by exchanging at most three vertices. We prove the following dichotomy: On the negative side, the problem of finding a 3-swap-optimal independent set or dominating set is PLS-complete. On the positive side, 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.
Local Search
Graph problems
3-swap neighborhood
PLS-completeness
Theory of computation~Problems, reductions and completeness
Theory of computation~Discrete optimization
Theory of computation~Graph algorithms analysis
66:1-66:14
Regular Paper
Christian
Komusiewicz
Christian Komusiewicz
Philipps-Universität Marburg, Fachbereich Mathematik und Informatik, Marburg, Germany
https://orcid.org/0000-0003-0829-7032
Nils
Morawietz
Nils Morawietz
Philipps-Universität Marburg, Fachbereich Mathematik und Informatik, Marburg, Germany
{Supported by the Deutsche Forschungsgemeinschaft (DFG), project OPERAH, KO 3669/5-1.}
10.4230/LIPIcs.MFCS.2022.66
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Christian Komusiewicz and Nils Morawietz
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