3-Coloring in Time O(1.3289^n)

TL;DR

This paper presents improved algorithms for NP-complete problems like 3-coloring by formulating them as CSPs and applying advanced backtracking, matching, and network flow techniques to achieve faster worst-case time bounds.

Contribution

It introduces a new fast algorithm for (3,2)-CSP and leverages it to improve time bounds for 3-SAT, 3-coloring, and related problems, using a CSP-based approach.

Findings

Achieved a time bound of O(1.3289^n) for 3-coloring.

Developed a fast (3,2)-CSP algorithm that underpins improvements.

Combined backtracking with matching and flow techniques for efficiency.

Abstract

We consider worst case time bounds for NP-complete problems including 3-SAT, 3-coloring, 3-edge-coloring, and 3-list-coloring. Our algorithms are based on a constraint satisfaction (CSP) formulation of these problems. 3-SAT is equivalent to (2,3)-CSP while the other problems above are special cases of (3,2)-CSP; there is also a natural duality transformation from (a,b)-CSP to (b,a)-CSP. We give a fast algorithm for (3,2)-CSP and use it to improve the time bounds for solving the other problems listed above. Our techniques involve a mixture of Davis-Putnam-style backtracking with more sophisticated matching and network flow based ideas.