Improved Algorithms for Counting Solutions in Constraint Satisfaction Problems

Abstract

Counting the number of solutions to CSP instances has vast applications in several areas ranging from statistical physics to artificial intelligence. We provide a new algorithm for counting the number of solutions to binary Csp s which has a time complexity ranging from $\mathcal{O}\left((d/4 \cdot \alpha^4)^n\right)$ to $\mathcal{O}\left((\alpha+\alpha^5 + \lfloor d/4-1 \rfloor \cdot \alpha^4)^n\right)$ (where α ≈ 1.2561) depending on the domain size d ≥ 3. This is substantially faster than previous algorithms, especially for small d. We also provide an algorithm for counting k-colourings in graphs and its running time ranges from $\mathcal{O}\left(\lfloor\log_2 k\rfloor^n\right)$ to $\mathcal{O}\left(\lfloor\log_2 k+1\rfloor^n\right)$ depending on k ≥ 4. Previously, only an $\mathcal{O}\left(1.8171^n\right)$ time algorithm for counting 3-colourings were known, and we improve this upper bound to $\mathcal{O}\left(1.7879^n\right)$.