We study the approximability of the MaxCut problem in the presence of predictions. Specifically, we consider two models: in the noisy predictions model, for each vertex we are given its correct label in $\{-1,+1\}$ with some unknown probability $1/2 + \epsilon$, and the other (incorrect) label otherwise. In the more-informative partial predictions model, for each vertex we are given its correct label with probability $\epsilon$ and no label otherwise. We assume only pairwise independence between vertices in both models. We show how these predictions can be used to improve on the worst-case approximation ratios for this problem. Specifically, we give an algorithm that achieves an $\alpha + \widetilde{\Omega}(\epsilon^4)$-approximation for the noisy predictions model, where $\alpha \approx 0.878$ is the MaxCut threshold. While this result also holds for the partial predictions model, we can also give a $\beta + \Omega(\epsilon)$-approximation, where $\beta \approx 0.858$ is the approximation ratio for MaxBisection given by Raghavendra and Tan. This answers a question posed by Ola Svensson in his plenary session talk at SODA'23.

翻译：我们研究在预测存在情况下MaxCut问题的可近似性。具体而言，我们考虑两种模型：在带噪预测模型中，每个顶点以未知概率$1/2 + \epsilon$获得其在$\{-1,+1\}$中的正确标签，否则获得错误标签；在信息更丰富的部分预测模型中，每个顶点以概率$\epsilon$获得正确标签，否则不获得标签。我们在两种模型中仅假设顶点之间的成对独立性。我们展示了如何利用这些预测来改进该问题的最坏情况近似比。具体地，我们给出了一种算法，在带噪预测模型中实现$\alpha + \widetilde{\Omega}(\epsilon^4)$-近似，其中$\alpha \approx 0.878$为MaxCut阈值。尽管该结果也适用于部分预测模型，我们还能给出$\beta + \Omega(\epsilon)$-近似，其中$\beta \approx 0.858$是Raghavendra和Tan提出的MaxBisection近似比。这回答了Ola Svensson在SODA'23全体会议演讲中提出的问题。