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Learning-augmented smooth integer programs with PAC-learnable oracles

Hao-Yuan He, Ming Li

TL;DR

This work develops a learning-augmented framework for smooth integer programs, notably MAX-CUT and MAX-k-SAT, by predicting a solution and linearizing the degree-d objective around that prediction to form an LP relaxation followed by rounding. The method achieves consistency (convergence to optimality as predictions improve), smoothness (graceful degradation with prediction error), and robustness (paralleling worst-case baselines), and extends dense PTAS guarantees to near-dense regimes. It provides a rigorous analysis of relaxation gaps and rounding for quadratic and general-degree polynomials, with explicit bounds that scale as powers of n and the error ε. Importantly, the paper establishes PAC-learnability of the oracle under bounded VC-dimension, via a pseudo-dimension-based uniform convergence argument, ensuring that near-optimal predictors can be learned with polynomial samples. Applications to MAX-CUT and MAX-k-SAT illustrate practical impact, and the use of full-information oracles eliminates sampling overhead present in prior work, broadening applicability to near-dense instances with provable guarantees.

Abstract

This paper investigates learning-augmented algorithms for smooth integer programs, covering canonical problems such as MAX-CUT and MAX-k-SAT. We introduce a framework that incorporates a predictive oracle to construct a linear surrogate of the objective, which is then solved via linear programming followed by a rounding procedure. Crucially, our framework ensures that the solution quality is both consistent and smooth against prediction errors. We demonstrate that this approach effectively extends tractable approximations from the classical dense regime to the near-dense regime. Furthermore, we go beyond the assumption of oracle existence by establishing its PAC-learnability. We prove that the induced algorithm class possesses a bounded pseudo-dimension, thereby ensuring that an oracle with near-optimal expected performance can be learned with polynomial samples.

Learning-augmented smooth integer programs with PAC-learnable oracles

TL;DR

This work develops a learning-augmented framework for smooth integer programs, notably MAX-CUT and MAX-k-SAT, by predicting a solution and linearizing the degree-d objective around that prediction to form an LP relaxation followed by rounding. The method achieves consistency (convergence to optimality as predictions improve), smoothness (graceful degradation with prediction error), and robustness (paralleling worst-case baselines), and extends dense PTAS guarantees to near-dense regimes. It provides a rigorous analysis of relaxation gaps and rounding for quadratic and general-degree polynomials, with explicit bounds that scale as powers of n and the error ε. Importantly, the paper establishes PAC-learnability of the oracle under bounded VC-dimension, via a pseudo-dimension-based uniform convergence argument, ensuring that near-optimal predictors can be learned with polynomial samples. Applications to MAX-CUT and MAX-k-SAT illustrate practical impact, and the use of full-information oracles eliminates sampling overhead present in prior work, broadening applicability to near-dense instances with provable guarantees.

Abstract

This paper investigates learning-augmented algorithms for smooth integer programs, covering canonical problems such as MAX-CUT and MAX-k-SAT. We introduce a framework that incorporates a predictive oracle to construct a linear surrogate of the objective, which is then solved via linear programming followed by a rounding procedure. Crucially, our framework ensures that the solution quality is both consistent and smooth against prediction errors. We demonstrate that this approach effectively extends tractable approximations from the classical dense regime to the near-dense regime. Furthermore, we go beyond the assumption of oracle existence by establishing its PAC-learnability. We prove that the induced algorithm class possesses a bounded pseudo-dimension, thereby ensuring that an oracle with near-optimal expected performance can be learned with polynomial samples.
Paper Structure (52 sections, 31 theorems, 96 equations, 4 algorithms)

This paper contains 52 sections, 31 theorems, 96 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Prior Works
  3. Our Contributions
  4. Organization
  5. Approach
  6. Preliminaries
  7. The Quadratic Case
  8. Oracle-guided Relaxation
  9. Analysis of the Relaxation Gap
  10. Randomized Rounding
  11. Comprehensive Guarantee
  12. Generalization to Higher Orders
  13. Oracle-guided Relaxation
  14. Analysis of the Relaxation Gap
  15. Rounding
  16. ...and 37 more sections

Key Result

Lemma 2.3

Let $p({\bm{x}})$ be an $n$-variate $\beta$-smooth polynomial of degree $d$. Then, there exist a constant $c$ and degree-$(d-1)$$\beta$-smooth polynomials $\{p_j({\bm{x}})\}_{j \in [n]}$ such that

Theorems & Definitions (59)

  • Definition 2.1: $\beta$-smooth
  • Example 1
  • Example 2
  • Definition 2.2: Problem Regimes
  • Lemma 2.3
  • Lemma 2.4
  • Corollary 2.4
  • Lemma 2.4: Lemma 3.2, Arora1999PTAS
  • Lemma 2.4
  • Remark 2.5
  • ...and 49 more