On exactness of SDP relaxation for the maximum cut problem

TL;DR

This work advances the theory of SDP relaxations for Max-Cut by (i) proving NP-hardness of recognizing exact GW-relaxation in unweighted graphs, (ii) identifying new unweighted graph families with exact relaxations and establishing explicit uniqueness criteria via a rank identity between primal and dual optima, and (iii) showing how exactness and solution rank propagate through graph operations such as the lexicographic product and split-decompositions. It demonstrates that a small structural core can enforce exactness in large graphs and addresses open questions from Mirka and Williamson by producing counterexamples and a complete uniqueness characterization for exact complete $k$-partite graphs. The results illuminate the geometry of the Max-Cut elliptope, reveal when higher-rank SDP optima occur, and suggest decomposition-based approaches to certify exactness in larger graphs. Overall, the paper contributes practical criteria and theoretical tools for understanding when SDP relaxations yield exact, or near-exact, solutions in diverse graph families. The work thus bridges combinatorial structure with semidefinite geometry to explain and leverage exactness beyond small instances.

Abstract

Semidefinite programming (SDP) provides a powerful relaxation for the maximum cut problem. For a graph with rational weights, the decision problem of whether the SDP relaxation for the maximum cut problem is exact is known to be NP-hard; however its complexity was unresolved for unweighted graphs. In this work, we extend the NP-hardness result to unweighted graphs. We characterize a few classes of graphs for which the SDP relaxation is exact. For each of these graph classes, we establish conditions for uniqueness of the SDP optimum. We complement these findings by identifying two graph operations that preserve the solution rank, and in turn exactness. These results reveal how the SDP relaxation for the maximum cut problem can remain exact in arbitrarily large graphs, owing to the presence of a small structural core that governs exactness. We further address two open problems posed by Mirka and Williamson (2024), by demonstrating that uniqueness of the maximum cut partition in exact relaxation does not imply uniqueness of the SDP optimum, and that exact relaxation with multiple optimal partitions may admit optimal SDP solutions lying outside the convex hull of rank-1 reference solutions.

Figures (2)

• Figure 1: Graph $\mathcal{G}$ with edge colors indicating weights: Red (1), Blue (2) & Green (6).
• Figure 2: Decomposition of $\mathcal{G}$. Edge colors indicate weights: Red (1), Blue (2), Green (6) & Black (18).

Theorems & Definitions (39)

• lemma 1
• theorem 1
• proof
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• proposition 3