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 $f:U\rightarrow \mathbb {R}\cup \{+\infty , -\infty \}$ 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.