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Parsimonious Learning-Augmented Approximations for Dense Instances of $\mathcal{NP}$-hard Problems

Evripidis Bampis, Bruno Escoffier, Michalis Xefteris

TL;DR

The paper addresses dense instances of $ ext{NP}$-hard optimization problems by introducing a learning-augmented approximation framework that uses a logarithmic number of one-bit predictions to speed up PTAS-like schemes. It develops LA-PTAS-Cut for Max-CUT, showing how sampling and predictions enable a linear-program formulation whose solution is efficiently rounded to a near-optimal cut; this yields $(1-ε-8 rac{ ext{error}}{δ|S|})$-type guarantees in time $O(nT_{LP})$ for $ ext{δ}$-dense graphs, and can be combined with Goemans–Williamson for robustness. The framework is generalized to smooth degree-$d$ polynomial programs (LA-PTAS, and LAA-General), enabling a broad method to obtain consistent and smooth additive approximations with running times independent of $1/ε$; these results extend to Max-$k$-SAT, Max-DICUT, Max-HYPERCUT($d$), and $k$-Densest Subgraph, illustrating practical speedups on dense instances while preserving worst-case guarantees. Overall, the work provides a scalable, prediction-aware approach to dense-instance optimization with theoretical guarantees, offering a path to faster, robust decision-making in algorithmic settings augmented by limited predictions.

Abstract

The classical work of (Arora et al., 1999) provides a scheme that gives, for any $ε>0$, a polynomial time $1-ε$ approximation algorithm for dense instances of a family of $\mathcal{NP}$-hard problems, such as Max-CUT and Max-$k$-SAT. In this paper we extend and speed up this scheme using a logarithmic number of one-bit predictions. We propose a learning augmented framework which aims at finding fast algorithms which guarantees approximation consistency, smoothness and robustness with respect to the prediction error. We provide such algorithms, which moreover use predictions parsimoniously, for dense instances of various optimization problems.

Parsimonious Learning-Augmented Approximations for Dense Instances of $\mathcal{NP}$-hard Problems

TL;DR

The paper addresses dense instances of -hard optimization problems by introducing a learning-augmented approximation framework that uses a logarithmic number of one-bit predictions to speed up PTAS-like schemes. It develops LA-PTAS-Cut for Max-CUT, showing how sampling and predictions enable a linear-program formulation whose solution is efficiently rounded to a near-optimal cut; this yields -type guarantees in time for -dense graphs, and can be combined with Goemans–Williamson for robustness. The framework is generalized to smooth degree- polynomial programs (LA-PTAS, and LAA-General), enabling a broad method to obtain consistent and smooth additive approximations with running times independent of ; these results extend to Max--SAT, Max-DICUT, Max-HYPERCUT(), and -Densest Subgraph, illustrating practical speedups on dense instances while preserving worst-case guarantees. Overall, the work provides a scalable, prediction-aware approach to dense-instance optimization with theoretical guarantees, offering a path to faster, robust decision-making in algorithmic settings augmented by limited predictions.

Abstract

The classical work of (Arora et al., 1999) provides a scheme that gives, for any , a polynomial time approximation algorithm for dense instances of a family of -hard problems, such as Max-CUT and Max--SAT. In this paper we extend and speed up this scheme using a logarithmic number of one-bit predictions. We propose a learning augmented framework which aims at finding fast algorithms which guarantees approximation consistency, smoothness and robustness with respect to the prediction error. We provide such algorithms, which moreover use predictions parsimoniously, for dense instances of various optimization problems.
Paper Structure (19 sections, 12 theorems, 27 equations, 2 algorithms)

This paper contains 19 sections, 12 theorems, 27 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Notation and Preliminaries
  4. Max-CUT
  5. Overview of Algorithm LA-PTAS-Cut
  6. Estimating Coefficients via Sampling and Predictions
  7. Formulating the Integer Linear Program
  8. Randomized Rounding
  9. Analysis of LA-PTAS-Cut
  10. LAA Framework
  11. Smooth Polynomial Integer Programs
  12. Applications
  13. Missing material of Section \ref{['general']}
  14. Integer Linear Program
  15. Estimating Polynomials via Sampling and Predictions
  16. ...and 4 more sections

Key Result

Theorem 1

Let $G$ a $\delta$-dense graph. Then, for any $\epsilon > 0$ with $|S|= \Theta(\ln n / (\epsilon^3 \delta^4))$, LAA-Cut runs in time $O(n \cdot T_{LP})$ and, with respect to the approximation ratio, is with high probability $(1-\epsilon)$-consistent, $(1-\epsilon-8\frac{\textit{error}}{\delta |S|})

Theorems & Definitions (16)

  • Theorem 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 6
  • Lemma 7
  • Theorem 8
  • Theorem 8
  • Lemma 9
  • ...and 6 more