2019-05-08 | Jingcheng Liu：A Deterministic Algorithm for Counting Colorings with 2? Colors
We give a polynomial time deterministic approximation algorithm (an FPTAS) for counting the number of q-colorings of a graph of maximum degree ?, provided only that q ≥ 2?. This substantially improves on previous deterministic algorithms for this problem, the best of which requires q ≥ 2.58?, 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 ≥ α? + β, where α ≈ 1.764 and β = β(α) 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.
Based on joint work with Alistair Sinclair and Piyush Srivastava.
Jingcheng Liu is a final year PhD candidate in the CS theory group at UC Berkeley, planning to graduate by Summer 2019 under the supervision of Professor Alistair Sinclair. Before coming to Berkeley, he did his undergraduate studies in Computer Science, in the ACM Honor Class 2010 at Shanghai Jiao Tong University. He is broadly interested in theoretical computer science. His current research focuses on the interplay between phase transitions in statistical physics, locations of zeros of graph polynomials, and algorithmic questions such as the tractable boundaries of approximate counting, sampling and inference.