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• W2898855866 abstract "Given a simple graph $G = (V, E)$ and a constant integer $k ge 2$, the $k$-path vertex cover problem ({sc P$k$VC}) 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 ({sc VC}) problem, which admits a $left(2 - {rm Theta}left(frac 1{log|V|}right)right)$-approximation. The general {sc P$k$VC} admits a trivial $k$-approximation; when $k = 3$ and $k = 4$, the best known approximation results for {sc P$3$VC} and {sc P$4$VC} are a $2$-approximation and a $3$-approximation, respectively. On $d$-regular graphs, the approximation ratios can be reduced to $minleft{2 - frac 5{d+3} + epsilon, 2 - frac {(2 - o(1))loglog d}{log d}right}$ for {sc VC} ({it i.e.}, {sc P$2$VC}), $2 - frac 1d + frac {4d - 2}{3d |V|}$ for {sc P$3$VC}, $frac {lfloor d/2rfloor (2d - 2)}{(lfloor d/2rfloor + 1) (d - 2)}$ for {sc P$4$VC}, and $frac {2d - k + 2}{d - k + 2}$ for {sc P$k$VC} when $1 le k-2 < d le 2(k-2)$. By utilizing an existing algorithm for graph defective coloring, we first present a $frac {lfloor d/2rfloor (2d - k + 2)}{(lfloor d/2rfloor + 1) (d - k + 2)}$-approximation for {sc P$k$VC} on $d$-regular graphs when $1 le k - 2 < d$. This beats all the best known approximation results for {sc P$k$VC} on $d$-regular graphs for $k ge 3$, except for {sc P$4$VC} 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 $1.875$-approximation and a $1.852$-approximation for {sc P$4$VC} on cubic graphs and $4$-regular graphs, respectively. We also present a better approximation algorithm for {sc P$4$VC} on $d$-regular bipartite graphs." @default.
• W2898855866 created "2018-11-09" @default.
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• W2898855866 date "2018-11-03" @default.
• W2898855866 modified "2023-10-17" @default.
• W2898855866 title "Improved approximation algorithms for path vertex covers in regular graphs" @default.
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• W2898855866 doi "https://doi.org/10.48550/arxiv.1811.01162" @default.
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