๐ค AI Summary
This work addresses the computational challenge of uniformly sampling proper $k$-colorings of graphs when the maximum degree $\Delta$ is large. The authors propose a novel approach based on partial rejection sampling (PRS), which introduces tunable soft coloring constraints that are progressively tightened to achieve exact uniform sampling. By integrating a recursive divide-and-conquer strategy, the original problem is decomposed into $O(\log n)$ independent subproblems of reduced size, each solvable in parallel by any exact sampler. This method is the first to combine soft coloring with PRS, enabling parallelization and achieving a runtime of $O(L^{\log^* n} \cdot n\Delta)$ when the number of relaxation levels $L$ is independent of $n$, improving upon the best-known algorithms that require $k > 3\Delta$. Empirical evidence suggests $L$ is likely constant, indicating potential for linear-time performance.
๐ Abstract
We present a new algorithm for the exact uniform sampling of proper \(k\)-colorings of a graph on \(n\) vertices with maximum degree~\(ฮ\). 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 \(k>3ฮ\). Recursive application of the hybrid drives the runtime to \(O(L^{\log^* n}\cdot nฮ)\), 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.