Lower Bounds of Functions on Finite Abelian Groups

TL;DR

This paper develops an efficient framework using semidefinite programming to compute lower bounds of functions on finite abelian groups, with applications to MAX-SAT problems and random functions, outperforming previous methods.

Contribution

It introduces a novel algorithm leveraging Fourier analysis and SDP to find verified lower bounds on functions over finite abelian groups, improving upon existing techniques.

Findings

Successfully applied to MAX-2SAT and MAX-3SAT benchmarks.

Demonstrated advantage over previous methods in experiments.

Effective on random functions on cyclic groups.

Abstract

The problem of computing the optimum of a function on a finite set is an important problem in mathematics and computer science. Many combinatorial problems such as MAX-SAT and MAXCUT can be recognized as optimization problems on the hypercube $C_2^n = \{-1,1\}^n$ consisting of $2^n$ points. It has been noticed that if a finite set is equipped with an abelian group structure, then one can efficiently certify nonnegative functions on it by Fourier sum of squares (FSOS). Motivated by these works, this paper is devoted to developing a framework to find a lower bound of a function on a finite abelian group efficiently. We implement our algorithm by the SDP solver SDPNAL+ for computing the verified lower bound of $f$ on $G$ and test it on the MAX-2SAT and MAX-3SAT benchmark problems from the Max-SAT competitions in 2009 and 2016. Beyond that, we also test our algorithm on random functions on $C_3^n$. These experiments demonstrate the advantage of our algorithm over previously known methods.