A Note on Max k-Vertex Cover: Faster FPT-AS, Smaller Approximate Kernel and Improved Approximation
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In Maximum k-Vertex Cover (Max k-VC), the input is an edge-weighted graph G and an integer k, and the goal is to find a subset S of k vertices that maximizes the total weight of edges covered by S. Here we say that an edge is covered by S iff at least one of its endpoints lies in S. We present an FPT approximation scheme (FPT-AS) that runs in (1/ϵ)^O(k) poly(n) time for the problem, which improves upon Gupta et al.'s (k/ϵ)^O(k) poly(n)-time FPT-AS [SODA'18, FOCS'18]. Our algorithm is simple: just use brute force to find the best k-vertex subset among the O(k/ϵ) vertices with maximum weighted degrees. Our algorithm naturally yields an efficient approximate kernelization scheme of O(k/ϵ) vertices; previously, an O(k^5/ϵ^2)-vertex approximate kernel is only known for the unweighted version of Max k-VC [Lokshtanov et al., STOC'17]. Interestingly, this has an application outside of parameterized complexity: using our approximate kernelization as a preprocessing step, we can directly apply Raghavendra and Tan's SDP-based algorithm for 2SAT with cardinality constraint [SODA'12] to give an 0.92-approximation for Max k-VC in polynomial time. This improves upon Feige and Langberg's algorithm [J. Algorithms'01] which yields (0.75 + δ)-approximation for some (unspecified) constant δ > 0. We also consider the minimization version (Min k-VC), where the goal is to minimize the total weight of edges covered by S. We provide an FPT-AS for Min k-VC with similar running time of (1/ϵ)^O(k) poly(n), which again improves on a (k/ϵ)^O(k) poly(n)-time FPT-AS of Gupta et al. On the other hand, we show that there is unlikely a polynomial size approximate kernelization for Min k-VC for any factor less than two.
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