Sharp Hardness for MAX-3-CUT and Quantum MAX-CUT

• URL: http://arxiv.org/abs/2608.00333v1
• Date: Fri, 31 Jul 2026 22:53:00 GMT
• Title: Sharp Hardness for MAX-3-CUT and Quantum MAX-CUT
• Abstract summary: We show it is NP-hard to approximate MAX-3-CUT within a multiplicative factor of $_3+$ for every $>0$.<n>With a similar strategy we prove it is NP-hard to approximate the product-state value of Quantum-CUT within a multiplicative factor of $_rm BOV+$ for every $>0$.
• Score: 2.538209532048867
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Assuming the Unique Games Conjecture, we show it is NP-hard to approximate MAX-3-CUT within a multiplicative factor of $α_3+ε$ for every $ε>0$, where $α_3\approx.83600811464$ is the approximation ratio of Frieze-Jerrum's polynomial-time algorithm from 1995. That is, we prove sharp hardness of approximation for MAX-3-CUT. This result resolves a conjecture of Khot-Kindler-Mossel-O'Donnell from 2004 by proving the three candidate Plurality is Stablest Conjecture for correlations in $[-1/2,2/5]$ and generalizes the Majority is Stablest Theorem of Mossel-O'Donnell-Oleszkiewicz [Annals of Math, 2010]. With a similar strategy we prove: assuming the Unique Games Conjecture, it is NP-hard to approximate the product-state value of Quantum MAX-CUT within a multiplicative factor of $α_{\rm BOV}+ε$ for every $ε>0$, where $α_{\rm BOV}\approx 0.9563372685$ is the approximation ratio of the Briët-de Oliveira Filho-Vallentin algorithm. This sharp hardness result completes the conjectured hardness of Hwang-Neeman-Parekh-Thompson-Wright from 2021 by proving their $S^{k-1}$-valued Borell inequality for correlations in $[-.5843,.5843]$ for all $k\geq3$.

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