Improved approximation algorithms for path vertex covers in regular graphs
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Given a simple graph G = (V, E) and a constant integer k > 2, the k-path vertex cover problem ( PkVC) asks for a minimum subset F ⊆ 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 (2 - Θ(1/|V|))-approximation. The general PkVC admits a trivial k-approximation; when k = 3 and k = 4, the best known approximation results for P3VC and P4VC are a 2-approximation and a 3-approximation, respectively. On d-regular graphs, the approximation ratios can be reduced to {2 - 5/d+3 + ϵ, 2 - (2 - o(1)) d/ d} for VC ( i.e., P2VC), 2 - 1/d + 4d - 2/3d |V| for P3VC, d/2 (2d - 2)/( d/2 + 1) (d - 2) for P4VC, and 2d - k + 2/d - k + 2 for PkVC when 1 < k-2 < d < 2(k-2). By utilizing an existing algorithm for graph defective coloring, we first present a d/2 (2d - k + 2)/( d/2 + 1) (d - k + 2)-approximation for PkVC on d-regular graphs when 1 < k - 2 < d. This beats all the best known approximation results for PkVC on d-regular graphs for 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-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.
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