Semidefinite programming bounds on fractional cut-cover and maximum 2-SAT for highly regular graphs
Henrique Assumpção, Gabriel Coutinho
TL;DR
This work develops semidefinite programming bounds for two graph parameters—fractional cut-cover $fcc$ and maximum 2-SAT—on graphs with strong symmetry, framed through antiblocking/gauge duality with $\eta(G)$ and $\eta^\circ(G)$. By leveraging coherent configurations and association schemes, it obtains exact or explicit spectral forms for key quantities: e.g., $\eta^\circ(G)=\dfrac{2k}{k-\lambda_{\min}(A)}$ for symmetric schemes, and $fcc$ bounds anchored by the Goemans–Williamson constant $\alpha_{GW}$. The authors prove that for graphs whose adjacency lies in or splits inside a coherent algebra, $\eta(G)\,\eta^\circ(G)=|E|$, extending equality cases to distance-regular and related graphs, and they derive LP representations to compute these quantities explicitly. They further generalize the SDP framework for approximating quadratic programs to the max 2-SAT setting, providing exact dual-optimal values for distance-regular graphs and highlighting cases where duality can be strict, which informs spectral bounds and potential approximation approaches with practical impact in combinatorial optimization on highly 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 \textsc{maxcut}, to graphs in certain coherent configurations. Moreover, we obtain spectral bounds for \textsc{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.