Sticky Brownian Rounding and its Applications to Constraint Satisfaction Problems
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Semidefinite programming is a powerful tool in the design and analysis of approximation algorithms for combinatorial optimization problems. In particular, the random hyperplane rounding method of Goemans and Williamson, has been extensively studied for more than two decades, resulting in various extensions to the original technique and beautiful algorithms for a wide range of applications. Despite the fact that this approach yields tight approximation guarantees for some problems, e.g., Max-Cut, for many others, e.g., Max-Sat and Max-DiCut, the tight approximation ratio is still unknown. One of the main reasons for this is the fact that very few techniques for rounding semi-definite relaxations are known. In this work, we present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired by recent results in algorithmic discrepancy theory. We develop and present tools for analyzing our new rounding algorithms, utilizing mathematical machinery from the theory of Brownian motion, complex analysis, and partial differential equations. Focusing on constraint satisfaction problems, we apply our method to several classical problems, including Max-Cut, Max-2SAT, and Max-DiCut, and derive new algorithms that are competitive with the best known results. We further show the versatility of our approach by presenting simple and natural variants of it, and we numerically demonstrate that they exhibit nearly optimal approximation guarantees for some problems.
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