Faster algorithm for Unique $(k,2)$-CSP

TL;DR

This paper presents improved algorithms for solving the Unique $(k,2)$-CSP problem, achieving faster running times for specific values of $k$, thus advancing the efficiency of solving these constraint satisfaction problems.

Contribution

The authors improve the running time bounds for Unique $(k,2)$-CSP for all $k \\geq 5$, notably enhancing previous algorithms for $k=5$ and $k=6$.

Findings

Improved running time for Unique (5,2)-CSP to $O(2.232^n)$

Enhanced bounds for Unique (6,2)-CSP to $O(2.641^n)$

General improvement for all $k \\geq 5$ in Unique $(k,2)$-CSP

Abstract

In a $(k,2)$-Constraint Satisfaction Problem we are given a set of arbitrary constraints on pairs of $k$-ary variables, and are asked to find an assignment of values to these variables such that all constraints are satisfied. The $(k,2)$-CSP problem generalizes problems like $k$-coloring and $k$-list-coloring. In the Unique $(k,2)$-CSP problem, we add the assumption that the input set of constraints has at most one satisfying assignment. Beigel and Eppstein gave an algorithm for $(k,2)$-CSP running in time $O\left(\left(0.4518k\right)^n\right)$ for $k>3$ and $O\left(1.356^n\right)$ for $k=3$, where $n$ is the number of variables. Feder and Motwani improved upon the Beigel-Eppstein algorithm for $k\geq 11$. Hertli, Hurbain, Millius, Moser, Scheder and Szedl{\'a}k improved these bounds for Unique $(k,2)$-CSP for every $k\geq 5$. We improve the result of Hertli et al. and obtain better bounds for Unique~$(k,2)$-CSP for~$k\geq 5$. In particular, we improve the running time of Unique~$(5,2)$-CSP from~$O\left(2.254^n\right)$ to~$O\left(2.232^n\right)$ and Unique~$(6,2)$-CSP from~$O\left(2.652^n\right)$ to~$O\left(2.641^n\right)$.