A deterministic algorithm for counting colorings with $2\Delta$ colors

Jingcheng Liu, Alistair Sinclair, Piyush Srivastava

We give a polynomial time deterministic approximation algorithm (an FPTAS) for counting the number of $q$-colorings of a graph of maximum degree $\Delta$, provided only that $q\ge 2\Delta$. This substantially improves on previous deterministic algorithms for this problem, the best of which requires $q\ge 2.58\Delta$, and matches the natural bound for randomized algorithms obtained by a straightforward application of Markov chain Monte Carlo. In the special case when the graph is also triangle-free, we show that our algorithm applies under the condition $q \geq \alpha\Delta + \beta$, where $\alpha \approx 1.764$ and $\beta = \beta(\alpha)$ are absolute constants. Our result applies more generally to list colorings, and to the partition function of the anti-ferromagnetic Potts model. Our algorithm exploits the so-called "polynomial interpolation" method of Barvinok, identifying a suitable region of the complex plane in which the Potts model partition function has no zeros. Interestingly, our method for identifying this zero-free region leverages probabilistic and combinatorial ideas that have been used in the analysis of Markov chains.

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