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Dependent k-Set Packing on Polynomoids

Authors Meng-Tsung Tsai , Shi-Chun Tsai , Tsung-Ta Wu

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Matroids and greedoids
Keywords
  • Hereditary Systems
  • Hypergraph Matchings
  • Compleixty Trichotomy

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Abstract

Specialized hereditary systems, e.g., matroids, are known to have many applications in algorithm design. We define a new notion called d-polynomoid as a hereditary system (E, ℱ ⊆ 2^E) so that every two maximal sets in ℱ have less than d elements in common. We study the problem that, given a d-polynomoid (E, ℱ), asks if the ground set E contains 𝓁 disjoint k-subsets that are not in ℱ, and obtain a complexity trichotomy result for all pairs of k ≥ 1 and d ≥ 0. Our algorithmic result yields a sufficient and necessary condition that decides whether each hypergraph in some classes of r-uniform hypergraphs has a perfect matching, which has a number of algorithmic applications.

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Meng-Tsung Tsai, Shi-Chun Tsai, and Tsung-Ta Wu. Dependent k-Set Packing on Polynomoids. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 84:1-84:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.MFCS.2023.84

Author Details

Meng-Tsung Tsai
  • Institute of Information Science, Academia Sinica, Taipei, Taiwan
Shi-Chun Tsai
  • Computer Science, National Yang Ming Chiao Tung University, Hsinchu, Taiwan
Tsung-Ta Wu
  • Computer Science, National Yang Ming Chiao Tung University, Hsinchu, Taiwan

Funding

This research was supported in part by the National Science and Technology Council under contract NSTC grants 108-2221-E-009-051-MY3, 109-2221-E-001-025-MY3, and 112-2221-E-001-007.

Acknowledgements

We sincerely thank anonymous reviewers for their helpful comments.

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