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

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.