Abstract
For a graph \(G=(V,E)\), a collection \(\mathcal {P}\) of vertex-disjoint (simple) paths is called a path cover of G if every vertex \(v\in V\) is contained in exactly one path of \(\mathcal {P}\). The Path Cover problem (PC for short) is to find a minimum cardinality path cover of G. In this paper, we introduce generalizations of PC, where each path is associated with a weight (cost or profit). Our problem, Minimum (Maximum) Weighted Path Cover [MinPC (MaxPC)], is defined as follows: Let \(U=\{0,1,\dots ,n-1\}\). Given a graph \(G=(V,E)\) and a weight function \(f:U\rightarrow \mathbb {R}\cup \{+\infty , -\infty \}\) that defines a weight for each path based on its length, the objective of MinPC (MaxPC) is to find a path cover \(\mathcal {P}\) of G such that the total weight of the paths in \(\mathcal {P}\) is minimized (maximized). Let L be a subset of U, and \(P^{L}\) be the set of paths such that each path is of length \(\ell \in L\). We consider Min\(P^{L}\) PC with binary cost, i.e., the cost function is \(f(\ell ) = 1\) if \(\ell \in L\); otherwise, \(f(\ell ) = 0\). We also consider Max\(P^{L}\) PC with \(f(\ell ) = \ell +1\), if \(\ell \in L\); otherwise, \(f(\ell ) = 0\). Many well-known graph theoretic problems such as the Hamiltonian Path and the Maximum Matching problems can be modeled using Min\(P^{L}\) PC and Max\(P^{L}\) PC. In this paper, we first show that deciding whether Min\(P^{\{0,1,2\}}\) PC has a 0-weight solution is NP-complete for planar bipartite graphs of maximum degree three, and consequently, (i) for any constant \(\sigma \ge 1\), there is no polynomial-time approximation algorithm with approximation ratio \(\sigma \) for Min\(P^{\{0,1,2\}}\) PC unless P \(=\) NP, and (ii) Max\(P^{\{3,\dots ,n-1\}}\) PC is NP-hard for the same graph class. Next, we present a polynomial-time algorithm for Min\(P^{\{0,1,\dots ,k\}}\) PC on graphs with bounded treewidth for a fixed k. Lastly, we present a 4-approximation algorithm for Max\(P^{\{3,\dots ,n-1\}}\) PC, which becomes a 2.5-approximation algorithm for subcubic graphs.
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Acknowledgements
The authors would like to thank the anonymous reviewers for their suggestions and detailed comments that helped to improve the presentation of the paper. This work is partially supported by NSERC Canada, JSPS KAKENHI Grant Numbers JP17K00016, JP18H04091, JP19K12098, JP20H05794, JP20K11666, JP21H05857 and JP21K11755, and JST CREST JPMJCR1402.
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A preliminary version of this paper appeared in Proceedings of the 16th International Conference and Workshops (WALCOM2022), LNCS 13174, pp.396–408, 2022.
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Kobayashi, K., Lin, G., Miyano, E. et al. Path Cover Problems with Length Cost. Algorithmica 85, 3348–3375 (2023). https://doi.org/10.1007/s00453-023-01106-2
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DOI: https://doi.org/10.1007/s00453-023-01106-2