There is a strong interest in finding challenging instances of NP-hard problems, from the perspective of showing quantum advantage. Due to the limits of near-term NISQ devices, it is moreover useful if these instances are small. In this work, we identify two graph families ($|V|<1000$) on which the Goemans-Williamson algorithm for approximating the Max-Cut achieves at most a 0.912-approximation. We further show that, in comparison, a recent quantum algorithm, Quantum Approximate Optimization Algorithm (depth $p=1$), is a 0.592-approximation on Karloff instances in the limit ($n \to \infty$), and is at best a $0.894$-approximation on a family of strongly-regular graphs. We further explore construction of challenging instances computationally by perturbing edge weights, which may be of independent interest, and include these in the CI-QuBe github repository.

翻译：从展示量子优势的角度来看，寻找NP难问题的挑战性实例具有强烈的兴趣。由于近期NISQ设备的限制，如果这些实例规模较小则更为有用。在本工作中，我们识别出两个图族（$|V|<1000$），在这些图上，用于近似求解最大割问题的Goemans-Williamson算法最多只能达到0.912的近似比。我们进一步表明，相比之下，一种近期提出的量子算法——量子近似优化算法（深度$p=1$）——在Karloff实例的极限情况下（$n \to \infty$）是一个0.592的近似算法，并且在一族强正则图上最多只能达到$0.894$的近似比。我们还通过计算方式探索了通过扰动边权重来构造挑战性实例的方法，这可能具有独立的研究价值，并将这些实例包含在CI-QuBe的github代码库中。