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.

翻译：我们提出了一种新算法，用于在具有 \(n\) 个顶点和最大度 \(Δ\) 的图上精确均匀采样其恰当的 \(k\)-着色。该算法基于部分拒绝采样（PRS），并通过引入恰当着色约束的软松弛形式逐步收紧，直至获得精确样本。与从过去耦合（CFTP）方法不同，该算法天然可并行化。我们进一步提出一种混合变体，将全局采样问题分解为若干独立子问题，每个子问题规模为 \(O(\log n)\)，可由任意现有精确求解器独立求解。这种分解充当 **复杂度缩减器**：它将组件求解器的输入规模从 \(n\) 替换为 \(O(\log n)\)，从而直接方法的任何改进均可自动转化为更强结果。以现有 CFTP 方法作为组件求解器时，该算法改进了已知最佳精确采样运行时间（针对 \(k>3Δ\) 的情况）。通过递归应用该混合方法，运行时间可降至 \(O(L^{\log^* n}\cdot nΔ)\)，其中 \(L\) 为松弛层数。我们推测 \(L\) 与 \(n\) 无关，若成立，则可为一般图提供线性时间可并行化算法。我们的仿真结果强烈支持这一猜想。