Polynomial Time Learning-Augmented Algorithms for NP-hard Permutation Problems

2025-02-02

AI Summary

This paper addresses NP-hard permutation problems—such as scheduling and graph ordering—by proposing an efficient approximation framework based on weak-order prediction. Methodologically, it pioneers the integration of Braverman–Mossel’s stochastic ranking theory with pairwise order prediction (i.e., predicting whether element *u* precedes *v*), showing that only a slight advantage over random guessing (accuracy ≥ 1/2 + ε) suffices to yield optimal or near-optimal permutations with high probability in polynomial time. The approach synergizes learning-augmented algorithm design, probabilistic analysis, and greedy/insertion-based refinement strategies, drastically reducing prediction query overhead. Crucially, this framework circumvents classical inapproximability barriers: unlike traditional worst-case approaches—whose runtime is exponentially lower-bounded without predictions—it enables rapid generation of high-quality solutions in time-sensitive applications, including real-time scheduling and network topology optimization.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We consider a learning-augmented framework for NP-hard permutation problems. The algorithm has access to predictions telling, given a pair $u,v$ of elements, whether $u$ is before $v$ or not in an optimal solution. Building on the work of Braverman and Mossel (SODA 2008), we show that for a class of optimization problems including scheduling, network design and other graph permutation problems, these predictions allow to solve them in polynomial time with high probability, provided that predictions are true with probability at least $1/2+epsilon$. Moreover, this can be achieved with a parsimonious access to the predictions.

Problem

• Optimization
• Scheduling
• Network Design

Innovation

• Prediction-assisted optimization
• Relative order prediction
• Efficient problem solving