Semidefinite programming bounds on fractional cut-cover and maximum 2-SAT for highly regular graphs

TL;DR

This paper develops semidefinite programming techniques to derive bounds on fractional cut-cover and MAX 2-SAT problems for highly regular graphs, extending known inequalities and computing optimal values in specific cases.

Contribution

It introduces new semidefinite programming bounds for fractional cut-cover and MAX 2-SAT in association schemes, and extends equality cases of primal-dual inequalities for MAXCUT.

Findings

Bound the fractional cut-cover parameter using eigenvalues.

Extend primal-dual inequality equality cases for MAXCUT.

Compute optimal values of gauge dual for distance-regular graphs.

Abstract

We use semidefinite programming to bound the fractional cut-cover parameter of graphs in association schemes in terms of their smallest eigenvalue. We also extend the equality cases of a primal-dual inequality involving the Goemans-Williamson semidefinite program, which approximates MAXCUT, to graphs in certain coherent configurations. Moreover, we obtain spectral bounds for MAX 2-SAT when the underlying graphs belong to a symmetric association scheme by means of a certain semidefinite program used to approximate quadratic programs, and we further develop this technique in order to explicitly compute the optimum value of its gauge dual in the case of distance-regular graphs.