An O(|E|)-linear Model for the MaxCut Problem

TL;DR

This paper introduces a new linear polytope model with at most 11|E| inequalities for the MaxCut problem, simplifying the problem to a polynomial-sized formulation especially for complete graphs.

Contribution

The paper presents a novel polytope model for MaxCut with at most 11|E| inequalities, each involving at most 4 variables, improving the linear formulation complexity.

Findings

Polytope $ ext{P}_{12}$ has at most $11|E|$ inequalities.

Each inequality involves at most 4 edge variables with coefficients $ ext{±}1$.

Model applies to complete graphs, reducing general MaxCut to this case.

Abstract

A polytope $P$ is a {\em model} for a combinatorial problem on finite graphs $G$ whose variables are indexed by the edge set $E$ of $G$ if the points of $P$ with (0,1)-coordinates are precisely the characteristic vectors of the subset of edges inducing the feasible configurations for the problem. In the case of the (simple) MaxCut Problem, which is the one that concern us here, the feasible subsets of edges are the ones inducing the bipartite subgraphs of $G$. In this paper we introduce a new polytope $\mathbb{P}_{12} \subset \mathbb{R}^{|E|}$ given by at most $11|E|$ inequalities, which is a model for the MaxCut Problem on $G$. Moreover, the left side of each inequality is the sum of at most 4 edge variables with coefficients $\pm1$ and right side 0,1, or 2. We restrict our analysis to the case of $G=K_{z}$, the complete graph in $z$ vertices, where $z$ is an even positive integer $z\ge 4$. This case is sufficient to study because the simple MaxCut problem for general graphs $G$ can be reduced to the complete graph $K_z$ by considering the obective function of the associated integer programming as the characteristic vector of the edges in $G \subseteq K_z$. This is a polynomial algorithmic transformation.