Layering and reverse extension for weighted $k$-path vertex cover

We study the weighted $k$-path vertex cover problem for any fixed integer $k \ge 2$, in which a vertex non-negatively weighted simple graph $G = (V, E)$ is given and one wants to find a minimum-weight vertex subset $C \subseteq V$ such that every simple path on $k$ vertices in $G$ contains at least one vertex of $C$. The problem generalizes the classic weighted vertex cover problem, that is, $k = 2$, and it admits a $k$-approximation algorithm of Hochbaum~[SIAM J. Comput., 1982] and a $(k - \frac 12)$-approximation algorithm of Brüstle et al.~[IWOCA 2021]. In this paper, we carefully design local weight functions in the Layering and Reverse Extension local ratio framework of Bar-Yehuda et al.~[ACM Comput. Surv., 2004], leading to an approximation algorithm of ratio $\frac {3k-1}4$ for odd $k \ge 3$ and of ratio $\frac {3k}4$ for even $k \ge 4$. When $k = 3$ and $k = 4$, these ratios are $2$ and $3$, respectively, matching the state-of-the-art approximation ratios for the corresponding special cases.