Improved Approximations for Hard Graph Problems using Predictions

TL;DR

This paper introduces new approximation algorithms for NP-hard graph problems that leverage predictions about optimal solutions, significantly improving performance even with weak predictive information.

Contribution

It extends the $ ext{epsilon}$-prediction framework to an edge-based model, enabling better approximation ratios for multiple graph problems using a unified approach.

Findings

Improved approximation ratios for MaxCut, Vertex Cover, Set Cover, and Independent Set.

Effective use of weak predictions to surpass traditional approximation barriers.

A unified algorithmic framework combining predictions for high-degree vertices and classical methods for low-degree vertices.

Abstract

We design improved approximation algorithms for NP-hard graph problems by incorporating predictions (e.g., learned from past data). Our prediction model builds upon and extends the $\varepsilon$-prediction framework by Cohen-Addad, d'Orsi, Gupta, Lee, and Panigrahi (NeurIPS 2024). We consider an edge-based version of this model, where each edge provides two bits of information, corresponding to predictions about whether each of its endpoints belong to an optimal solution. Even with weak predictions where each bit is only $\varepsilon$-correlated with the true solution, this information allows us to break approximation barriers in the standard setting. We develop algorithms with improved approximation ratios for MaxCut, Vertex Cover, Set Cover, and Maximum Independent Set problems (among others). Across these problems, our algorithms share a unifying theme, where we separately satisfy constraints related to high degree vertices (using predictions) and low-degree vertices (without using predictions) and carefully combine the answers.