← 研踪 PaperTrailarXiv:2611.03115 · v1
Faster Exact Algorithms for 3- and 4-Coloring: Deterministic Improvements and Further Randomized Speedups
首次提交 2026-11-02 21:32
版本更新 2026-11-02 21:32
雷达观测 2026-11-04 23:41
摘要
We study exact exponential-time algorithms for 3- and 4-coloring. For 3-coloring, the best previously published running-time bound is $O^*(1.3289^n)$, due to Eppstein (SODA 2001). We improve this bound to $O^*(1.3204^n)$ deterministically and $O^*(1.3159^n)$ using randomization. For 4-coloring, the best previously known running-time bound is $O^*(1.7159^n)$, due to Wu, Gu, Jiang, Shao, and Xu (ESA 2024). We improve this bound to $O^*(1.6975^n)$ deterministically and $O^*(1.6933^n)$ using randomization. Our algorithms for both problems combine constraint satisfaction algorithms with dynamic programming over tree and path decompositions, exploiting their complementary strengths on different graph structures. For 6-coloring, the best published randomized running-time bound is exponentially smaller than the best published deterministic bound. For 3- and 4-coloring, however, no such improvement through randomization was previously known. To the best of our knowledge, our results provide the first randomized running-time bounds for these two problems that exponentially improve upon the best previously published deterministic bounds. This raises the question of whether randomization offers an inherent exponential advantage for 3- and 4-coloring.
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v1 · 2026-11-02 21:32 · 最新观测版本
Faster Exact Algorithms for 3- and 4-Coloring: Deterministic Improvements and Further Randomized Speedups
We study exact exponential-time algorithms for 3- and 4-coloring. For 3-coloring, the best previously published running-time bound is $O^*(1.3289^n)$, due to Eppstein (SODA 2001). We improve this bound to $O^*(1.3204^n)$ deterministically and $O^*(1.3159^n)$ using randomization. For 4-coloring, the best previously known running-time bound is $O^*(1.7159^n)$, due to Wu, Gu, Jiang, Shao, and Xu (ESA 2024). We improve this bound to $O^*(1.6975^n)$ deterministically and $O^*(1.6933^n)$ using randomization. Our algorithms for both problems combine constraint satisfaction algorithms with dynamic programming over tree and path decompositions, exploiting their complementary strengths on different graph structures. For 6-coloring, the best published randomized running-time bound is exponentially smaller than the best published deterministic bound. For 3- and 4-coloring, however, no such improvement through randomization was previously known. To the best of our knowledge, our results provide the first randomized running-time bounds for these two problems that exponentially improve upon the best previously published deterministic bounds. This raises the question of whether randomization offers an inherent exponential advantage for 3- and 4-coloring.
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