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Maximum Coverage k-Antichains and Chains: A Greedy Approach

Authors Manuel Cáceres , Andreas Grigorjew , Wanchote Po Jiamjitrak , Alexandru I. Tomescu

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LIPIcs.ESA.2026.108.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Network flows
  • Theory of computation → Sorting and searching
Keywords
  • Maximum coverage antichains
  • maximum coverage chains
  • directed acyclic graph
  • minimum cost flow
  • greedy set cover
  • parameterized algorithms
  • approximation algorithms

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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 α_k (resp. β_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 simplify and improve previous results. Specifically, we obtain the following for MA-k:  
- An algorithm running in |E|^{1+o(1)} time, and an algorithm running in parameterized near-linear Õ(α_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(α₁²|V| + (α₁+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.  Additionally, we obtain analogous results for MC-k as well as the corresponding dual problems derived from the Greene-Kleitman theorems, which might be of independent interest.
We complement these results with two examples (one for chains and one for antichains) showing that, for every k ≥ 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 Ω(log{|V|}) factor away from minimality when required to cover all vertices: previously unknown for sets of chains or antichains.

Cite As Get BibTex

Manuel Cáceres, Andreas Grigorjew, Wanchote Po Jiamjitrak, and Alexandru I. Tomescu. Maximum Coverage k-Antichains and Chains: A Greedy Approach. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 108:1-108:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.108

Author Details

Manuel Cáceres
  • Department of Computer Science, Aalto University, Espoo, Finland
  • Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark
Andreas Grigorjew
  • LAMSADE, CNRS UMR7243, Université Paris Dauphine-PSL, 75775 Paris, France
  • Institute of Informatics, University of Warsaw, Poland
Wanchote Po Jiamjitrak
  • Department of Computer Science, University of Helsinki, Finland
Alexandru I. Tomescu
  • Department of Computer Science, University of Helsinki, Finland

Funding

Co-funded by the European Union (ERC, SCALEBIO, 101169716). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. Co-funded also by the Research Council of Finland grants No. 346968, 358744.
  • Cáceres, Manuel: Supported by the Helsinki Institute for Information Technology HIIT.
  • Grigorjew, Andreas: Supported by ANR project ANR-21-CE48-0022 (S-EX-AP-PE-AL).

Acknowledgements

We appreciate the insightful comments provided by the anonymous reviewers.

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