MAX-3-CUT与量子MAX-CUT的强难解性
Sharp Hardness for MAX-3-CUT and Quantum MAX-CUT
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中文总结 AI 辅助
假设唯一游戏猜想，该研究证明MAX-3-CUT和量子MAX-CUT的近似问题分别在对应因子下是NP难的，解决了相关猜想并推广了稳定定理。
中文摘要 AI 辅助
假设唯一游戏猜想成立，我们证明对于任意ε>0，MAX-3-CUT在乘法因子α₃+ε下近似是NP难的，其中α₃≈0.83600811464是Frieze-Jerrum 1995年提出的多项式时间算法的近似比，即证明了MAX-3-CUT的强近似难解性。该结果通过证明[-1/2,2/5]范围内的相关性对应的三个候选“多数是稳定的猜想”，解决了Khot-Kindler-Mossel-O'Donnell 2004年的猜想，并推广了Mossel-O'Donnell-Oleszkiewicz（《数学年刊》，2010）的“少数是稳定的定理”。采用类似策略，我们证明假设唯一游戏猜想成立，对于任意ε>0，量子MAX-CUT的乘积态值在乘法因子α_BOV+ε下近似是NP难的，其中α_BOV≈0.9563372685是Briët-de Oliveira Filho-Vallentin算法的近似比。该强难解性结果通过证明k≥3时[-0.5843,0.5843]范围内的相关性对应的Hwang-Neeman-Parekh-Thompson-Wright 2021年猜想的S^{k-1}值Borell不等式，完善了该猜想的难解性。
英文摘要
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 $k\geq3$.
