Computation with Polynomial Equations and Inequalities arising in Combinatorial Optimization

TL;DR

This paper surveys algebraic methods using polynomial equations and inequalities to address combinatorial optimization problems through large-scale linear algebra and semidefinite programming relaxations.

Contribution

It introduces a methodology leveraging polynomial algebra to formulate and solve combinatorial optimization problems more effectively.

Findings

Effective relaxations for combinatorial problems

Application of semidefinite programming techniques

Framework for solving polynomial systems in optimization

Abstract

The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear algebra or semidefinite programming relaxations of many kinds of feasibility or optimization questions. We are particularly interested in problems arising in combinatorial optimization.