Improved Approximation Algorithms for Path Vertex Covers in Regular Graphs

Abstract

Given a simple graph $$G = (V, E)$$ and a constant integer $$k \ge 2$$ , the k-path vertex cover problem (PkVC) asks for a minimum subset $$F \subseteq V$$ of vertices such that the induced subgraph $$G[V - F]$$ does not contain any path of order k. When $$k = 2$$ , this turns out to be the classic vertex cover (VC) problem, which admits a $$\left( 2 - {\Theta }\left( \frac{1}{\log |V|}\right) \right)$$ -approximation. The general PkVC admits a trivial k-approximation; when $$k = 3$$ and $$k = 4$$ , the best known approximation results are a 2-approximation and a 3-approximation, respectively. On d-regular graphs, the approximation ratios can be reduced to $$\min \left\{ 2 - \frac{5}{d+3} + \epsilon , 2 - \frac{(2 - o(1))\log \log d}{\log d}\right\}$$ for VC (i.e., P2VC), $$2 - \frac{1}{d} + \frac{4d - 2}{3d |V|}$$ for P3VC, $$\frac{\lfloor d/2\rfloor (2d - 2)}{(\lfloor d/2\rfloor + 1) (d - 2)}$$ for P4VC, and $$\frac{2d - k + 2}{d - k + 2}$$ for PkVC when $$1 \le k-2 < d \le 2(k-2)$$ . By utilizing an existing algorithm for graph defective coloring, we first present a $$\frac{\lfloor d/2\rfloor (2d - k + 2)}{(\lfloor d/2\rfloor + 1) (d - k + 2)}$$ -approximation for PkVC on d-regular graphs when $$1 \le k - 2 < d$$ . This beats all the best known approximation results for PkVC on d-regular graphs for $$k \ge 3$$ , except for P4VC it ties with the best prior work and in particular they tie at 2 on cubic graphs and 4-regular graphs. We then propose a $$(1.875 + \epsilon )$$ -approximation and a 1.852-approximation for P4VC on cubic graphs and 4-regular graphs, respectively. We also present a better approximation algorithm for P4VC on d-regular bipartite graphs.