The Complexity of Recognizing SDP Exactness for the Maximum Cut Problem

Avinash Bhardwaj

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arXiv · 2609.03508

The Complexity of Recognizing SDP Exactness for the Maximum Cut Problem

Avinash Bhardwaj

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Abstract

The standard semidefinite programming (SDP) relaxation of Max-Cut is exact when its optimum equals the maximum cut value. Delorme and Poljak proved NP-completeness of recognizing exactness for weighted graphs and left the unweighted case open. We prove that recognizing exactness is NP-complete even for connected simple unweighted graphs, and hence strongly NP-complete for graphs with nonnegative integer weights. The reduction represents the clauses of a linear monotone NAE-4-SAT instance by edge-disjoint copies of $K_6$. An explicit vector assignment attains a common SDP bound, and the additive integrality gap equals the minimum number of unsatisfied clauses. Hardness persists for simple unweighted graphs when an exact rational optimal primal--dual pair is supplied. Shared simplex anchors establish strong NP-hardness of recognizing exactness of the Frieze--Jerrum Max-$k$-Cut SDP relaxation for every fixed $k\ge3$, even for connected graphs with nonnegative integer edge weights. We also give an independent bounded-weight sum-of-squares proof of strong NP-completeness of recognizing exactness of the Max-Cut SDP, together with gap-preserving reductions establishing strong NP-completeness of exactness recognition for a basic Max-DiCut SDP and NP-hardness for a Max-Bisection SDP.

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Avinash Bhardwaj. 2026-09-03. The Complexity of Recognizing SDP Exactness for the Maximum Cut Problem. https://arxiv.org/abs/2609.03508

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