Max-Cut with $ε$-Accurate Predictions
Vincent Cohen-Addad, Tommaso d'Orsi, Anupam Gupta, Euiwoong Lee, Debmalya Panigrahi
TL;DR
This work studies MaxCut under two prediction-based models: noisy predictions where each vertex's label is correct with probability $\frac{1}{2}+\varepsilon$, and partial predictions where a vertex label is revealed with probability $\varepsilon$, both under pairwise independence. It develops a framework that leverages predictions via Δ-wide/Δ-narrow graph decompositions, LP relaxations, and specialized rounding (Goemans–Williamson, RT12) to surpass the classical GW threshold $\alpha_{GW}$, achieving $\alpha_{GW}+\tilde{Ω}(\varepsilon^4)$ in the noisy model and $\alpha_{RT}+Ω(ε)$ in the partial model. The methods extend to dense 2-CSPs, yielding PTAS-like behavior in dense instances and establishing a path toward leveraging predictions to beat worst-case bounds in beyond-worst-case settings. The results address a question posed by Svensson at SODA'23 and provide practical algorithmic templates for prediction-augmented combinatorial optimization with potential impact on ML-assisted optimization pipelines.
Abstract
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 $1/2 + ε$, and the other (incorrect) label otherwise. In the more-informative partial predictions model, for each vertex we are given its correct label with probability $ε$ 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 $α+ \widetildeΩ(ε^4)$-approximation for the noisy predictions model, where $α\approx 0.878$ is the MaxCut threshold. While this result also holds for the partial predictions model, we can also give a $β+ Ω(ε)$-approximation, where $β\approx 0.858$ 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.