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arXiv · 2610.34324

Optimal solution ranks in the Max-Cut semidefinite relaxation

Avinash Bhardwaj·Vishnu Narayanan
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Abstract

For every $n$ and $r$ with $1\le r$ and $\binom{r+1}{2}\le n$, we construct a connected unweighted graph on $n$ vertices whose Max-Cut semidefinite relaxation has a unique optimal solution of rank $r$. The main examples are the Kneser graphs $K(m,2)$, whose unique optima have rank $m-1$. A six-vertex wheel supplies rank three, and attaching pendant vertices preserves both rank and uniqueness. For dense random graphs, we also prove a polynomial lower bound on the optimal rank.

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Avinash Bhardwaj, Vishnu Narayanan. 2026-10-25. Optimal solution ranks in the Max-Cut semidefinite relaxation. https://arxiv.org/abs/2610.34324

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