CASP: Learning-Augmented Offline Approximation with Verifiable Certificates and Bounded-Loss PAC Guarantees

2026-07-16

AI Summary

This work addresses the challenge of accelerating offline NP-hard optimization using machine learning while preserving worst-case correctness guarantees. The authors propose CASP, a framework that integrates learned predictions with polynomial-time verifiable certificates: a predicted solution is accepted only when accompanied by a valid certificate, ensuring correctness independent of prediction quality. CASP introduces a verifiable pruning mechanism that enables distribution-agnostic probably approximately correct (PAC) learnability and enhances solver efficiency by breaking symmetries in degenerate solution spaces. Experiments across five NP-hard problems demonstrate that unverified pruning can incur up to 26% optimality loss under distribution shift, whereas CASP achieves zero performance degradation while maintaining rigorous theoretical guarantees.

Abstract

Machine-learned predictions can speed up offline NP-hard optimization, but asking a predictor what to do amounts to asking it to solve the problem, and committing an unchecked prediction forfeits every worst-case guarantee. CASP (Certificate-Augmented Solution Pruning) instead asks which parts of the search space may be ignored, and accepts each answer only after a sound polynomial-time verifier has checked it, so correctness never depends on prediction quality. We develop the learning theory of this design. The verifier makes the induced loss class uniformly bounded, so certificate parameters are learnable from $\tilde O(\varepsilon^{-2}\log K)$ samples ($K$ the maximum instance size), whereas the unverified commitment class admits no distribution-free rate and, under cost spread $R$, none below $Ω(R/\varepsilon^2)$. Filtering noisy predictions by verifiable confidence dominates the standard min-combiner, with a margin we compute in closed form, and the prediction stays useful even given the LP, because it breaks ties on degenerate optimal faces, where every symmetric LP policy, meaning one whose commitments depend on the instance only through the verifiable confidence values, provably stalls. Experiments on five problems test the theory's quantitative predictions. With trained predictors, unverified pruning loses up to $26%$ of the optimum under distribution shift, while the verified deployment of the same predictions loses nothing.

Problem

• offline optimization
• NP-hard problems
• machine-learned predictions
• worst-case guarantees
• verifiable certificates

Innovation

• Certificate-Augmented Solution Pruning
• Verifiable Certificates
• Bounded-Loss PAC Guarantees
• Learning-Augmented Algorithms
• Offline Optimization