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 \(k\)-colorings of a graph on \(n\) vertices with maximum degree~\(\Delta\). 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 \(k>3\Delta\). Recursive application of the hybrid drives the runtime to \(O(L^{\log^* n}\cdot n\Delta)\), 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.