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In a $(k,2)$-Constraint Satisfaction Problem we are given a set of arbitraryconstraints on pairs of $k$-ary variables, and are asked to find an assignmentof 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 assumptionthat 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 improvedupon 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 forevery $k\geq 5$.  We improve the result of Hertli et al. and obtain better bounds forUnique~$(k,2)$-CSP for~$k\geq 5$. In particular, we improve the running time ofUnique~$(5,2)$-CSP from~$O\left(2.254^n\right)$ to~$O\left(2.232^n\right)$ andUnique~$(6,2)$-CSP from~$O\left(2.652^n\right)$ to~$O\left(2.641^n\right)$.

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