Perfect Sampling for (Atomic) Lovász Local Lemma
吴克文，University of California at Berkeley
2021-07-12 15:00:00 ~ 2021-07-12 16:30:00
We give a Markov chain based perfect sampler for uniform sampling solutions of constraint satisfaction problems (CSP). Under some mild Lovász local lemma conditions where each constraint of the CSP has a small number of forbidden local configurations, our algorithm is accurate and efficient: it outputs a perfect uniform random solution and its expected running time is quasilinear in the number of variables. Prior to our work, perfect samplers are only shown to exist for CSPs under much more restrictive conditions (Guo, Jerrum, and Liu, JACM’19).
Our algorithm has two components:
• A simple perfect sampling algorithm using bounding chains (Huber, STOC’98; Haggstrom and Nelander, Scandinavian Journal of Statistics’99).
This sampler is efficient if each variable domain is small.
• A simple but powerful state tensorization trick to reduce large domains to smaller ones.
This trick is a generalization of state compression (Feng, He, and Yin, STOC’21).
The crux of our analysis is a simple information percolation argument which allows us to achieve bounds even beyond current best approximate samplers (Jain, Pham, and Vuong, ArXiv’21).
Previous related works either use intricate algorithms or needs sophisticated analysis or even both. Thus we view the simplicity of both our algorithm and analysis as a strength of our work.
Joint work with Kun He and Xiaoming Sun.
Kewen Wu is a first-year graduate student, supervised by Prof. Avishay Tal, in the theory group of University of California at Berkeley.
Before joining UC Berkeley, he received Bachelor's degree in Computer Science and Math from Peking University.
He has broad interest in theoretical computer science and its connection with mathematics.