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arXiv · 2502.06459

Maximum Coverage $k$-Antichains and Chains: A Greedy Approach

Manuel Cáceres·Andreas Grigorjew·Wanchote Po Jiamjitrak·Alexandru I. Tomescu
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Abstract

Given an acyclic digraph $G = (V,E)$ and a positive integer $k$, the problem of Maximum Coverage $k$-Antichains (resp. Chains) denoted as MA-$k$ (resp. MC-$k$) asks to find $k$ sets of pairwise unreachable vertices, known as antichains (resp. $k$ subsequences of paths, known as chains), maximizing the number $\alpha_k$ (resp. $\beta_k$) of vertices covered by these antichains (resp. chains). While MC-$k$ was solved in almost optimal $|E|^{1+o(1)}$ time~[Kogan and Parter, ICALP'22], the fastest algorithms for MA-$k$ are a $(k|E|)^{1+o(1)}$-time solution and a $|E|^{1+o(1)}$-time $1/2$ approximation~[Kogan and Parter, ESA'24]. We obtain the following for MA-$k$: - An algorithm running in $|E|^{1+o(1)}$ time, and an algorithm running in parameterized near-linear $\tilde{O}(\alpha_k |E|)$ time. Our algorithms are simple solutions exploiting a paths-based proof of the Greene-Kleitman theorems leveraged by the greedy algorithm for set cover as well as recent advances in fast algorithms for flows and shortest paths. - An approximation algorithm running in parameterized linear time $O(\alpha_1^2|V| + (\alpha_1+k)|E|)$ with approximation ratio of $(1-1/e) > 0.63 > 1/2$, beating the state-of-the-art $1/2$ approximation. Our solution uses greedy for antichains and a simple strategy to amortize the cost of computing consecutive maximum antichains. We complement these results with two examples (one for chains and one for antichains) showing that, for every $k \ge 2$, greedy misses the tight $1/e$ portion of the optimal coverage for chains, and a $1/4$ portion for antichains. We also show that greedy is a $\Omega(\log{|V|})$ factor away from minimality when required to cover all vertices: previously unknown for sets of chains or antichains.

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Manuel Cáceres, Andreas Grigorjew, Wanchote Po Jiamjitrak, Alexandru I. Tomescu. 2025-02-10. Maximum Coverage $k$-Antichains and Chains: A Greedy Approach. https://arxiv.org/abs/2502.06459

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