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On lower bounds of the density of planar periodic sets without unit distances

Alexander Tolmachev

TL;DR

The paper investigates the maximal density $m_1(\mathbb{R}^2)$ of planar 1-avoiding sets by restricting to planar sets that are periodic with respect to two non-collinear vectors and reformulating the problem as a maximum independent set (MIS) problem on graphs defined on a flat torus. It introduces a hexagonal partition of a perfectly periodic flat torus $T_{l_1,l_2,\alpha}$ and constructs corresponding graphs whose independent sets yield lower bounds $m_1(\mathbb{R}^2) \ge |M|/(nm)$. Across a spectrum of torus parameters and MIS solvers, the best results align with Croft's long-standing construction, with lower bounds around $0.22936$ not being improved within the tested regime; KaMIS typically provides the strongest MIS, while neural methods offer competitive performance in some regimes but face scaling challenges. The work demonstrates the viability of MIS-based periodic searches for $m_1(\mathbb{R}^2)$ and provides open-source tools and benchmarks for related geometric-graph problems.

Abstract

Determining the maximal density $m_1(\mathbb{R}^2)$ of planar sets without unit distances is a fundamental problem in combinatorial geometry. This paper investigates lower bounds for this quantity. We introduce a novel approach to estimating $m_1(\mathbb{R}^2)$ by reformulating the problem as a Maximal Independent Set (MIS) problem on graphs constructed from flat torus, focusing on periodic sets with respect to two non-collinear vectors. Our experimental results, supported by theoretical justifications of proposed method, demonstrate that for a sufficiently wide range of parameters this approach does not improve the known lower bound $0.22936 \le m_1(\mathbb{R}^2)$. The best discrete sets found are approximations of Croft's construction. In addition, several open source software packages for MIS problem are compared on this task.

On lower bounds of the density of planar periodic sets without unit distances

TL;DR

The paper investigates the maximal density of planar 1-avoiding sets by restricting to planar sets that are periodic with respect to two non-collinear vectors and reformulating the problem as a maximum independent set (MIS) problem on graphs defined on a flat torus. It introduces a hexagonal partition of a perfectly periodic flat torus and constructs corresponding graphs whose independent sets yield lower bounds . Across a spectrum of torus parameters and MIS solvers, the best results align with Croft's long-standing construction, with lower bounds around not being improved within the tested regime; KaMIS typically provides the strongest MIS, while neural methods offer competitive performance in some regimes but face scaling challenges. The work demonstrates the viability of MIS-based periodic searches for and provides open-source tools and benchmarks for related geometric-graph problems.

Abstract

Determining the maximal density of planar sets without unit distances is a fundamental problem in combinatorial geometry. This paper investigates lower bounds for this quantity. We introduce a novel approach to estimating by reformulating the problem as a Maximal Independent Set (MIS) problem on graphs constructed from flat torus, focusing on periodic sets with respect to two non-collinear vectors. Our experimental results, supported by theoretical justifications of proposed method, demonstrate that for a sufficiently wide range of parameters this approach does not improve the known lower bound . The best discrete sets found are approximations of Croft's construction. In addition, several open source software packages for MIS problem are compared on this task.

Paper Structure

This paper contains 11 sections, 6 theorems, 25 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Periodic grids
  3. Graph construction
  4. Flat torus partitioning into equal hexagons
  5. Experiments & Results
  6. Experiment scheme
  7. Independent set solvers and machine learning for combinatorial optimization
  8. Evaluation and comparison of various solvers on flat torus based graphs
  9. Analysis of experimental results
  10. Conclusion
  11. Acknowledgments.

Key Result

lemma thmcounterlemma

Let $T_{l_1, l_2, \alpha}$ be the flat torus such that $l_1 \le l_2$. Let $p_1 = (x_1, y_1)$, $p_2 = (x_2, y_2) \in T_{l_1, l_2,\alpha}$ be arbitrary points. Then,

Figures (9)

  • Figure 1: Construction for the proof of Lemma \ref{['lemma_perfectly_periodic']} in case $y_2 - y_1 \in [0, 1)$.
  • Figure 2: Graph construction concepts.
  • Figure 3: The construction of the graph $G_{n, m}$ in case of $n = 5, m = 4$. Dark hexagons denote the independent set $M$. This example leads to the following estimation: $m_1(\mathbb{R}^2) \ge \frac{|M|}{nm} = 4 / 20 = 0.2$.
  • Figure 4: KaMIS results for $G_{n, n}$ graphs based on torus $T_{l_1^*, l_2^*, \alpha^*}$
  • Figure 5: DGL-TreeSearch results for $G_{n, n}$ graphs based on torus $T_{l_1^*, l_2^*, \alpha^*}$
  • ...and 4 more figures

Theorems & Definitions (17)

  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • proof
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • ...and 7 more