Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniform Sampling of Proper Graph Colorings via Soft Coloring and Partial Rejection Sampling

Sarat Moka, Ava Vahedi

Abstract

We present a new algorithm for the exact uniform sampling of proper \(k\)-colorings of a graph on \(n\) vertices with maximum degree~\(Δ\). The algorithm is based on partial rejection sampling (PRS) and introduces a soft relaxation of the proper coloring constraint that is progressively tightened until an exact sample is obtained. Unlike coupling from the past (CFTP), the method is inherently parallelizable. We propose a hybrid variant that decomposes the global sampling problem into independent subproblems of size \(O(\log n)\), each solved by any existing exact sampler. This decomposition acts as a {\em complexity reducer}: it replaces the input size~\(n\) with \(O(\log n)\) in the component solver's runtime, so that any improvement in direct methods automatically yields a stronger result. Using an existing CFTP method as the component solver, this improves upon the best known exact sampling runtime for \(k>3Δ\). Recursive application of the hybrid drives the runtime to \(O(L^{\log^* n}\cdot nΔ)\), where \(L\) is the number of relaxation levels. We conjecture that \(L\) is bounded independently of~\(n\), which would yield a linear-time parallelizable algorithm for general graphs. Our simulations strongly support this conjecture.

Uniform Sampling of Proper Graph Colorings via Soft Coloring and Partial Rejection Sampling

Abstract

We present a new algorithm for the exact uniform sampling of proper -colorings of a graph on vertices with maximum degree~. The algorithm is based on partial rejection sampling (PRS) and introduces a soft relaxation of the proper coloring constraint that is progressively tightened until an exact sample is obtained. Unlike coupling from the past (CFTP), the method is inherently parallelizable. We propose a hybrid variant that decomposes the global sampling problem into independent subproblems of size \(O(\log n)\), each solved by any existing exact sampler. This decomposition acts as a {\em complexity reducer}: it replaces the input size~ with \(O(\log n)\) in the component solver's runtime, so that any improvement in direct methods automatically yields a stronger result. Using an existing CFTP method as the component solver, this improves upon the best known exact sampling runtime for . Recursive application of the hybrid drives the runtime to \(O(L^{\log^* n}\cdot nΔ)\), where is the number of relaxation levels. We conjecture that is bounded independently of~, which would yield a linear-time parallelizable algorithm for general graphs. Our simulations strongly support this conjecture.

Paper Structure

This paper contains 20 sections, 11 theorems, 39 equations, 9 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partial Rejection Sampling
  4. Perfect Sampling for Graph Colorings
  5. gamma-Soft Coloring
  6. The New Algorithm
  7. Sampling of TEXT-Soft Coloring
  8. Parallelization
  9. Hybrid $\gamma$-PRS
  10. Runtime Analysis
  11. Probability of a Bad Vertex
  12. Expected Number of Bad Vertices
  13. Non-Degeneration Condition: the Percolation Threshold
  14. Sufficient Condition on $k$
  15. Runtime of the Hybrid Algorithm
  16. ...and 5 more sections

Key Result

Theorem 1

For any graph, the distribution of $\gamma$-soft coloring converges to the uniform distribution on proper colorings as $\gamma$ goes to zero. That is, for any sample $\boldsymbol{x}$, $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (30)

  • Example 1: Hard-Core Model
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Remark 1
  • Proposition 1
  • proof
  • ...and 20 more