A Symmetry-Based Explanation of the Main Idea Behind Chubanov’s Linear Programming Algorithm

Abstract

Many important real-life optimization problems can be described as optimizing a linear objective function under linear constraints—i.e., as a linear programming problem. This problem is known to be not easy to solve. Reasonably natural algorithms—such as iterative constraint satisfaction or simplex method—often require exponential time. There exist efficient polynomial-time algorithms, but these algorithms are complicated and not very intuitive. Also, in contrast to many practical problems which can be computed faster by using parallel computers, linear programming has been proven to be the most difficult to parallelize. Recently, Sergei Chubanov proposed a modification of the iterative constraint satisfaction algorithm: namely, instead of using the original constraints, he proposed to come up with appropriate derivative constraints. Interestingly, this idea leads to a new polynomial-time algorithm for linear programming—and to efficient algorithms for many other constraint satisfaction problems. In this paper, we show that an algebraic approach—namely, the analysis of the corresponding symmetries—can (at least partially) explain the empirical success of Chubanov’s idea.