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Matching in Geometric Uniform Hypergraphs

Authors Matthew J. Katz , Yuval Nidam , Rachel Saban , Micha Sharir

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LIPIcs.ESA.2026.48.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Geometric hypergraphs
  • maximum matching
  • PTAS
  • sparsification
  • local search

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Abstract

Let P be a set of n points in ℝ^d, d ≥ 2, and let t ≥ 2 be an integer. Let H_t(P) denote the t-uniform hypergraph on P, whose hyperedges consist of all t-tuples T ⊂ P for which ‖p-q‖ ≤ 1, for any two points p,q ∈ T. A matching in H_t(P) is a collection of vertex-disjoint hyperedges. We present a PTAS for finding a maximum matching in H_t(P). In particular, we present the first PTAS for the well-studied problem known as maximum (vertex-disjoint) triangle packing in unit disk graphs. Our approach consists of a sparsification stage, which replaces P by a subset Q with favorable properties, followed by an implementation of a PTAS for a maximum matching in H_t(Q). The two stages follow the high-level machinery in [Édouard Bonnet et al., 2023] and [Rom Aschner et al., 2013], respectively, but are considerably more involved.

Cite As Get BibTex

Matthew J. Katz, Yuval Nidam, Rachel Saban, and Micha Sharir. Matching in Geometric Uniform Hypergraphs. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 48:1-48:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.48

Author Details

Matthew J. Katz
  • The Stein Faculty of Computer and Information Science, Ben-Gurion University of the Negev, Israel
Yuval Nidam
  • The Stein Faculty of Computer and Information Science, Ben-Gurion University of the Negev, Israel
Rachel Saban
  • The Stein Faculty of Computer and Information Science, Ben-Gurion University of the Negev, Israel
Micha Sharir
  • School of Computer Science, Tel Aviv University, Israel

Funding

  • Katz, Matthew J.: Partially supported by Israel Science Foundation Grant 495/23.
  • Sharir, Micha: Partially supported by Israel Science Foundation Grant 495/23.

References

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