📝 SummarXiv

Abstract

Semidefinite programming (SDP) provides a powerful relaxation of the maximum cut problem. For a graph $\mathcal{G}$ with rational weights, the decision problem of whether this relaxation is exact is known to be $NP$-complete since Delorme-Poljak (1993), but its complexity was unresolved for unweighted graphs. In this work, we extend the $NP$-completeness result for unweighted graphs. We then investigate and present a few classes of graphs, which we call exact graphs, for which the semidefinite relaxation of the maximum cut problem is exact. For each exact graph class, we determine the optimal objective value by explicitly constructing a maximum cut and then prove uniqueness of the exact solution for two of these classes. We complement these findings by showing that lexicographic products preserve exactness whenever the first factor is exact, with rank-$r$ solutions extending accordingly, and that a graph is exact if and only if when all of its splits are exact, with splitting also preserving the rank of optimal solutions. Addressing two open problems posed by Mirka and Williamson (2024), we demonstrate via counterexamples that exact semidefinite relaxations can admit higher-rank optimal solutions beyond the convex hull of rank-1 reference solutions. We then explain this phenomenon by proving the existence of rank-$(n-1)$ optimal solutions for particular classes of graphs. Finally, we give necessary and sufficient conditions under which the corresponding solution to exact relaxation is unique for particular classes of complete graphs and complete $k$-partite graphs.