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Proper colorings of a graph in linear time using a number of colors linear in the maximum degree of the graph

Kritika Bhandari, Mark Huber

TL;DR

This work tackles the problem of exactly sampling a uniform proper $k$-coloring of a graph with maximum degree $\Delta$. It introduces a randomness recycler (RR) protocol tailored to colorings, encoding per-vertex constraints as an index $x^*$ and progressively removing restrictions via three operations to reach an all-$\bot$ target, while preserving the exact target distribution. The main result proves that if $k > 3.637\Delta + 1$, the expected running time is linear in the number of vertices, with each step costing $O(\Delta)$ and requiring $O(\Delta \log k)$ random bits. This achieves linear-time exact sampling for colorings in the regime where the color count scales linearly with $\Delta$, improving prior bounds and enabling practical exact sampling for graphs with moderate to large degree.

Abstract

A new algorithm for exactly sampling from the set of proper colorings of a graph is presented. This is the first such algorithm that has an expected running time that is guaranteed to be linear in the size of a graph with maximum degree \( Δ\) when the number of colors is greater than \( 3.637 Δ+ 1\).

Proper colorings of a graph in linear time using a number of colors linear in the maximum degree of the graph

TL;DR

This work tackles the problem of exactly sampling a uniform proper -coloring of a graph with maximum degree . It introduces a randomness recycler (RR) protocol tailored to colorings, encoding per-vertex constraints as an index and progressively removing restrictions via three operations to reach an all- target, while preserving the exact target distribution. The main result proves that if , the expected running time is linear in the number of vertices, with each step costing and requiring random bits. This achieves linear-time exact sampling for colorings in the regime where the color count scales linearly with , improving prior bounds and enabling practical exact sampling for graphs with moderate to large degree.

Abstract

A new algorithm for exactly sampling from the set of proper colorings of a graph is presented. This is the first such algorithm that has an expected running time that is guaranteed to be linear in the size of a graph with maximum degree when the number of colors is greater than .
Paper Structure (8 sections, 3 theorems, 46 equations)

This paper contains 8 sections, 3 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. The algorithm
  3. The Randomness Recycler Protocol
  4. The RR algorithm for proper colorings
  5. Remove ignored
  6. Remove frozen
  7. Remove forbidden
  8. Proving Theorem 1

Key Result

Theorem 1

There exists a randomness recycler algorithm for generating a sample that is an exact uniform draw from the proper colorings of a graph with $n$ nodes and maximum degree $\Delta$ that uses (on average) $\Theta(n)$ steps each taking $O(\Delta)$ time to execute and using at most $O(\Delta \log(k))$ un

Theorems & Definitions (5)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof