Given a simple graph $$G = (V, E)$$ G = ( V , E ) and a constant integer $$k \ge 2$$ k ≥ 2 , the k -path vertex cover problem ( P k VC ) asks for a minimum subset $$F \subseteq V$$ F ⊆ V of vertices such that the induced subgraph $$G[V - F]$$ G [ V - F ] does not contain any path of order k . When $$k = 2$$ k = 2 , this turns out to be the classic vertex cover ( VC ) problem, which admits a $$\left( 2 - {\Theta }\left( \frac{1}{\log |V|}\right) \right)$$ 2 - Θ 1 log | V | -approximation. The general P k VC admits a trivial k -approximation; when $$k = 3$$ k = 3 and $$k = 4$$ 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 $$\min \left\{ 2 - \frac{5}{d+3} + \epsilon , 2 - \frac{(2 - o(1))\log \log d}{\log d}\right\}$$ min 2 - 5 d + 3 + ϵ , 2 - ( 2 - o ( 1 ) ) log log d log d for VC (i.e., P2VC ), $$2 - \frac{1}{d} + \frac{4d - 2}{3d |V|}$$ 2 - 1 d + 4 d - 2 3 d | V | for P3VC , $$\frac{\lfloor d/2\rfloor (2d - 2)}{(\lfloor d/2\rfloor + 1) (d - 2)}$$ ⌊ d / 2 ⌋ ( 2 d - 2 ) ( ⌊ d / 2 ⌋ + 1 ) ( d - 2 ) for P4VC , and $$\frac{2d - k + 2}{d - k + 2}$$ 2 d - k + 2 d - k + 2 for P k VC when $$1 \le k-2 < d \le 2(k-2)$$ 1 ≤ k - 2 < d ≤ 2 ( k - 2 ) . By utilizing an existing algorithm for graph defective coloring, we first present a $$\frac{\lfloor d/2\rfloor (2d - k + 2)}{(\lfloor d/2\rfloor + 1) (d - k + 2)}$$ ⌊ d / 2 ⌋ ( 2 d - k + 2 ) ( ⌊ d / 2 ⌋ + 1 ) ( d - k + 2 ) -approximation for P k VC on d -regular graphs when $$1 \le k - 2 < d$$ 1 ≤ k - 2 < d . This beats all the best known approximation results for P k VC on d -regular graphs for $$k \ge 3$$ k ≥ 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 $$(1.875 + \epsilon )$$ ( 1.875 + ϵ ) -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.

中文翻译：

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正则图中路径顶点覆盖的改进逼近算法

给定一个简单的图 $$G = (V, E)$$ G = ( V , E ) 和一个常数整数 $$k \ge 2$$ k ≥ 2 ，k 路径顶点覆盖问题 ( P k VC )要求顶点的最小子集 $$F \subseteq V$$ F ⊆ V 使得诱导子图 $$G[V - F]$$ G [ V - F ] 不包含任何 k 阶路径。当 $$k = 2$$ k = 2 时，这证明是经典的顶点覆盖 (VC) 问题，它承认 $$\left( 2 - {\Theta }\left( \frac{1}{\ log |V|}\right) \right)$$ 2 - Θ 1 log | V | -近似。一般的 P k VC 承认一个平凡的 k -近似；当 $$k = 3$$ k = 3 和 $$k = 4$$ k = 4 时，最著名的近似结果分别是 2-近似和 3-近似。在 d -正则图上，近似比率可以减少到 $$\min \left\{ 2 - \frac{5}{d+3} + \epsilon ，2 - \frac{(2 - o(1))\log \log d}{\log d}\right\}$$ min 2 - 5 d + 3 + ϵ , 2 - ( 2 - o ( 1 ) ) log log d log d 用于 VC（即 P2VC），$$2 - \frac{1}{d} + \frac{4d - 2}{3d |V|}$$ 2 - 1 d + 4 d - 2 3 d | V | 对于 P3VC , $$\frac{\lfloor d/2\rfloor (2d - 2)}{(\lfloor d/2\rfloor + 1) (d - 2)}$$ ⌊ d / 2 ⌋ ( 2 d - 2 ) ( ⌊ d / 2 ⌋ + 1 ) ( d - 2 ) 对于 P4VC 和 $$\frac{2d - k + 2}{d - k + 2}$$ 2 d - k + 2 d - k + 2 对于 P k VC，当 $$1 \le k-2 < d \le 2(k-2)$$ 1 ≤ k - 2 < d ≤ 2 ( k - 2 ) 。通过利用现有的图缺陷着色算法，我们首先提出 $$\frac{\lfloor d/2\rfloor (2d - k + 2)}{(\lfloor d/2\rfloor + 1) (d - k + 2)}$$ ⌊ d / 2 ⌋ ( 2 d - k + 2 ) ( ⌊ d / 2 ⌋ + 1 ) ( d - k + 2 ) - d -正则图上的 P k VC 近似值，当 $$1 \ le k - 2 < d$$ 1 ≤ k - 2 < d 。这在 $$k \ge 3$$ k ≥ 3 的 d 正则图上击败了 P k VC 的所有最著名的近似结果，除了 P4VC 它与最好的先前工作相关，特别是它们在三次图上与 2 相关联和 4 个正则图。然后，我们分别为三次图和 4 正则图上的 P4VC 提出了 $$(1.875 + \epsilon )$$ ( 1.875 + ϵ ) 近似值和 1.852 近似值。我们还在 d 正则二部图上提出了一种更好的 P4VC 近似算法。