Path Cover Problems with Length Cost

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 $f:U\rightarrow \mathbb {R}\cup \{+\infty , -\infty \}$, which defines a weight for each path in its length, 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 especially consider $\textsf {Min}P^{L}\textsf {PC}$ with 0–1 cost, i.e., the cost function is $f(\ell ) = 1$ if $\ell \in L$; otherwise $f(\ell ) = 0$. We also consider $\textsf {Max}P^{L}\textsf {PC}$ with $f(\ell ) = \ell +1$, if $\ell \in L$; otherwise $f(\ell ) = 0$. That is, $\textsf {Max}P^{L}\textsf {PC}$ is to maximize the number of vertices contained in the paths with length $\ell \in L$ in a path cover. In this paper, we first show that $\textsf {Min}P^{\{0,1,2\}}\textsf {PC}$ is NP-hard for planar bipartite graphs of maximum degree three. This implies that (i) for any constant $\sigma \ge 1$, there is no polynomial-time approximation algorithm with approximation ratio $\sigma$ for $\textsf {Min}P^{\{0,1,2\}}\textsf {PC}$ unless P$=$NP, and (ii) $\textsf {Max}P^{\{3,\dots ,n-1\}}\textsf {PC}$ is NP-hard for the same graph class. Next, (iii) we present a polynomial-time algorithm for $\textsf {Min}P^{\{0,1,\dots ,k\}}\textsf {PC}$ on graphs with bounded treewidth for a fixed k. Lastly, (iv) we present a 4-approximation algorithm for $\textsf {Max}P^{\{3,\dots ,n-1\}}\textsf {PC}$, which becomes a 2.5-approximation for subcubic graphs.