AI 中文总结
该研究打破了图k着色平均复杂度的二次壁垒，证明其平均复杂度为Θ(nk)且最优，还得到了探针复杂度为poly(k)的局部计算算法。
AI 中文摘要
图k着色是经典的NP完全问题之一。前人研究了其平均时间复杂度，定义为在所有n个顶点的k可着色图集合上计算k着色的平均运行时间。1989年Dyer-Frieze的极具影响力的结果给出了常数k下平均运行时间为O(n²)的算法，该二次运行时间看似自然（甚至可能最优），因为几乎所有k可着色图都有Θ(n²)条边，因此至少需要这么多时间才能读取整个输入。然而，1995年Kučera将其改进为对于每个k ≤ n^c（其中c∈(0,1)），平均运行时间为O(n²/k)。不过，在最受关注的k=O(1)的情况下，已知的最佳界限仍为n的二次方。过去三十年来，k着色问题的真实平均复杂度一直难以捉摸。我们打破了长期存在的二次壁垒，本文的主要结果表明，对于每个k ≤ n^{c'}（其中c'∈(0,1)），这一基础问题的精确平均情况复杂度为Θ(nk)。对于k=O(1)，这揭示了k着色性的平均次线性性质：平均情况复杂度线性于n，因此在输入规模上是次线性的。我们进一步证明，我们的Θ(nk)平均运行时间是最优的，因为一个简单的界限表明，所有能正确对所有k可着色图进行k着色的算法，都需要Ω(nk)的平均运行时间。我们的证明借鉴了次线性算法和局部算法的思想，还产生了一种k着色的局部计算算法（LCA），其平均情况探针复杂度为poly(k)。我们算法中的一个关键新要素是利用图正则性理论的工具来证明随机子图的唯一着色性的方法。
英文摘要
Graph $k$-coloring is one of the classic NP-complete problems. Previous work has studied its average time complexity, defined to be the average runtime of computing a $k$-coloring over the set of all $k$-colorable graphs on $n$ vertices. A highly influential result of Dyer-Frieze from 1989 gave an algorithm with $O(n^2)$ average runtime for constant $k$. This quadratic runtime appeared natural (and possibly even optimal) since almost all $k$-colorable graphs have $Θ(n^2)$ edges, so one needs at least this time in order to read the (entire) input. However, this was later improved by Kučera in 1995 to average runtime $O(n^2/k)$ for every $k \leq n^{c}$ where $c \in (0, 1)$. Nevertheless, in the most interesting case of $k = O(1)$, the best-known bound remained quadratic in $n$. The true average complexity of the $k$-coloring problem has remained elusive for the last three decades. We break the longstanding quadratic barrier. Our main result in this paper shows that the exact average-case complexity of this fundamental problem is $Θ(nk)$ for every $k \leq n^{c'}$ and some $c' \in (0, 1)$. For $k = O(1)$, this reveals the average sublinear nature of $k$-colorability: the average-case complexity is linear in $n$, and thus sublinear in the size of the input. We further show that our $Θ(nk)$ average runtime is optimal, since a simple bound proves that every algorithm that correctly $k$-colors all $k$-colorable graphs requires $Ω(n k)$ average runtime. Our proofs draw on ideas from sublinear and local algorithms and also yield a local computation algorithm (LCA) for $k$-coloring with average-case probe complexity $\text{poly}(k)$. A key new ingredient in our algorithm is a method for certifying the unique colorability of random subgraphs, using tools from the theory of graph regularity.
Comments40 pages, 5 figures. FOCS 2026
