Abstract
We study the capacitated vertex cover problem (CVC). In this natural extension to the vertex cover problem, each vertex has a predefined capacity which indicates the total amount of edges that it can cover. In this paper, we study the complexity of the CVC problem. We give NP-completeness proofs for the problem on modular graphs, tree-convex graphs, and planar bipartite graphs of maximum degree three. For the first two graph classes, we prove that no subexponential-time algorithm exist for CVC unless the ETH fails.
Furthermore, we introduce a series of exact exponential-time algorithms which solve the CVC problem on several graph classes in \(\mathcal {O}((2 - \epsilon )^n)\) time, for some \(\epsilon > 0\). Amongst these graph classes are, graphs of maximum degree three, other degree-bounded graphs, regular graphs, graphs with large matchings, c-sparse graphs, and c-dense graphs. To obtain these results, we introduce an FPT treewidth algorithm which runs in \(\mathcal {O}^*((k + 1)^{tw})\) or \(\mathcal {O}^*(k^k)\) time, where k is the solution size and tw the treewidth, improving an earlier algorithm from Dom et al.
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Notes
- 1.
Note that Vertex Cover equals Independet Set and Clique when considered from the viewpoint of exact exponential-time algorithms.
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van Rooij, S.B., van Rooij, J.M.M. (2019). Algorithms and Complexity Results for the Capacitated Vertex Cover Problem. In: Catania, B., Královič, R., Nawrocki, J., Pighizzini, G. (eds) SOFSEM 2019: Theory and Practice of Computer Science. SOFSEM 2019. Lecture Notes in Computer Science(), vol 11376. Springer, Cham. https://doi.org/10.1007/978-3-030-10801-4_37
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