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• W2897300971 abstract "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 $(1/epsilon)^{O(k)} poly(n)$ time for the problem, which improves upon Gupta et al.'s $(k/epsilon)^{O(k)} poly(n)$-time FPT-AS [SODA'18, FOCS'18]. Our algorithm is simple: just use brute force to find the best $k$-vertex subset among the $O(k/epsilon)$ vertices with maximum weighted degrees. Our algorithm naturally yields an efficient approximate kernelization scheme of $O(k/epsilon)$ vertices; previously, an $O(k^5/epsilon^2)$-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 $(0.75 + delta)$-approximation for some (unspecified) constant $delta > 0$. 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 $(1/epsilon)^{O(k)} poly(n)$, which again improves on a $(k/epsilon)^{O(k)} poly(n)$-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." @default.
• W2897300971 created "2018-10-26" @default.
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• W2897300971 date "2018-10-09" @default.
• W2897300971 modified "2023-09-27" @default.
• W2897300971 title "A Note on Max $k$-Vertex Cover: Faster FPT-AS, Smaller Approximate Kernel and Improved Approximation" @default.
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