Improved Approximation Algorithms for Path Vertex Covers in Regular Graphs

Given a simple graph G = (V, E) and a constant integer k ≥ 2 , the k -path vertex cover problem ( P k VC ) 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/log |V|) ) -approximation. The general P k VC 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{ 2 - 5/d+3 + ϵ , 2 - (2 - o(1))loglog d/log 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 P k VC 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 P k VC on d -regular graphs when 1 ≤ k - 2 < d . This beats all the best known approximation results for P k VC 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 + 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.