For a graph G=(V,E)\documentclass[12pt]{minimal}
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\begin{document}$$G=(V,E)$$\end{document}, a collection P\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {P}$$\end{document} of vertex-disjoint (simple) paths is called a path cover of G if every vertex v∈V\documentclass[12pt]{minimal}
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\begin{document}$$v\in V$$\end{document} is contained in exactly one path of P\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {P}$$\end{document}. 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,⋯,n-1}\documentclass[12pt]{minimal}
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\begin{document}$$U=\{0,1,\dots ,n-1\}$$\end{document}. Given a graph G=(V,E)\documentclass[12pt]{minimal}
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\begin{document}$$G=(V,E)$$\end{document} and a weight function f:U→R∪{+∞,-∞}\documentclass[12pt]{minimal}
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\begin{document}$$f:U\rightarrow \mathbb {R}\cup \{+\infty , -\infty \}$$\end{document} that defines a weight for each path based on its length, the objective of MinPC (MaxPC) is to find a path cover P\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {P}$$\end{document} of G such that the total weight of the paths in P\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {P}$$\end{document} is minimized (maximized). Let L be a subset of U, and PL\documentclass[12pt]{minimal}
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\begin{document}$$P^{L}$$\end{document} be the set of paths such that each path is of length ℓ∈L\documentclass[12pt]{minimal}
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\begin{document}$$\ell \in L$$\end{document}. We consider MinPL\documentclass[12pt]{minimal}
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\begin{document}$$P^{L}$$\end{document}PC with binary cost, i.e., the cost function is f(ℓ)=1\documentclass[12pt]{minimal}
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\begin{document}$$f(\ell ) = 1$$\end{document} if ℓ∈L\documentclass[12pt]{minimal}
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\begin{document}$$\ell \in L$$\end{document}; otherwise, f(ℓ)=0\documentclass[12pt]{minimal}
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\begin{document}$$f(\ell ) = 0$$\end{document}. We also consider MaxPL\documentclass[12pt]{minimal}
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\begin{document}$$P^{L}$$\end{document}PC with f(ℓ)=ℓ+1\documentclass[12pt]{minimal}
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\begin{document}$$f(\ell ) = \ell +1$$\end{document}, if ℓ∈L\documentclass[12pt]{minimal}
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\begin{document}$$\ell \in L$$\end{document}; otherwise, f(ℓ)=0\documentclass[12pt]{minimal}
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\begin{document}$$f(\ell ) = 0$$\end{document}. Many well-known graph theoretic problems such as the Hamiltonian Path and the Maximum Matching problems can be modeled using MinPL\documentclass[12pt]{minimal}
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\begin{document}$$P^{L}$$\end{document}PC and MaxPL\documentclass[12pt]{minimal}
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\begin{document}$$P^{L}$$\end{document}PC. In this paper, we first show that deciding whether MinP{0,1,2}\documentclass[12pt]{minimal}
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\begin{document}$$P^{\{0,1,2\}}$$\end{document}PC has a 0-weight solution is NP-complete for planar bipartite graphs of maximum degree three, and consequently, (i) for any constant σ≥1\documentclass[12pt]{minimal}
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\begin{document}$$\sigma \ge 1$$\end{document}, there is no polynomial-time approximation algorithm with approximation ratio σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document} for MinP{0,1,2}\documentclass[12pt]{minimal}
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\begin{document}$$P^{\{0,1,2\}}$$\end{document}PC unless P =\documentclass[12pt]{minimal}
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\begin{document}$$=$$\end{document} NP, and (ii) MaxP{3,⋯,n-1}\documentclass[12pt]{minimal}
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\begin{document}$$P^{\{3,\dots ,n-1\}}$$\end{document}PC is NP-hard for the same graph class. Next, we present a polynomial-time algorithm for MinP{0,1,⋯,k}\documentclass[12pt]{minimal}
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\begin{document}$$P^{\{0,1,\dots ,k\}}$$\end{document}PC on graphs with bounded treewidth for a fixed k. Lastly, we present a 4-approximation algorithm for MaxP{3,⋯,n-1}\documentclass[12pt]{minimal}
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\begin{document}$$P^{\{3,\dots ,n-1\}}$$\end{document}PC, which becomes a 2.5-approximation algorithm for subcubic graphs.
