Convex Hulls for Graphs of Quadratic Functions With Unit Coefficients: Even Wheels and Complete Split Graphs

TL;DR

This paper develops extended formulations for the convex hull of quadratic functions defined on specific graph classes, such as even wheels and complete split graphs, using Boolean quadric polytope facets.

Contribution

It provides new extended formulations for the convex hull of quadratic functions on even wheel and complete split graphs, expanding understanding of their polyhedral structure.

Findings

Extended formulations for even wheel graphs

Extended formulations for complete split graphs

Characterization of convex hulls using Boolean quadric polytope facets

Abstract

We study the convex hull of the graph of a quadratic function $f(\mathbf{x})=\sum_{ij\in E}x_ix_j$, where the sum is over the edge set of a graph $G$ with vertex set $\{1,\dots,n\}$. Using an approach proposed by Gupte et al. (Discrete Optimization $\textbf{36}$, 2020, 100569), we investigate minimal extended formulations using additional variables $y_{ij}$, $1\leq i<j\leq n$, representing the products $x_ix_j$. The basic idea is to identify a set of facets of the Boolean Quadric Polytope which is sufficient for characterizing the convex hull for the given graph. Our main results are extended formulations for the cases that the underlying graph $G$ is either an even wheel or a complete split graph.