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Approximation Algorithms for Combinatorial Optimization with Predictions

Antonios Antoniadis, Marek Eliáš, Adam Polak, Moritz Venzin

TL;DR

There is a potential for exploiting specific structural properties of individual problems to obtain improved bounds on the Steiner Tree problem and the approach is shown to be optimal for this class of problems as a whole.

Abstract

We initiate a systematic study of utilizing predictions to improve over approximation guarantees of classic algorithms, without increasing the running time. We propose a systematic method for a wide class of optimization problems that ask to select a feasible subset of input items of minimal (or maximal) total weight. This gives simple (near-)linear time algorithms for, e.g., Vertex Cover, Steiner Tree, Min-Weight Perfect Matching, Knapsack, and Clique. Our algorithms produce optimal solutions when provided with perfect predictions and their approximation ratios smoothly degrade with increasing prediction error. With small enough prediction error we achieve approximation guarantees that are beyond reach without predictions in the given time bounds, as exemplified by the NP-hardness and APX-hardness of many of the above problems. Although we show our approach to be optimal for this class of problems as a whole, there is a potential for exploiting specific structural properties of individual problems to obtain improved bounds; we demonstrate this on the Steiner Tree problem. We conclude with an empirical evaluation of our approach.

Approximation Algorithms for Combinatorial Optimization with Predictions

TL;DR

There is a potential for exploiting specific structural properties of individual problems to obtain improved bounds on the Steiner Tree problem and the approach is shown to be optimal for this class of problems as a whole.

Abstract

We initiate a systematic study of utilizing predictions to improve over approximation guarantees of classic algorithms, without increasing the running time. We propose a systematic method for a wide class of optimization problems that ask to select a feasible subset of input items of minimal (or maximal) total weight. This gives simple (near-)linear time algorithms for, e.g., Vertex Cover, Steiner Tree, Min-Weight Perfect Matching, Knapsack, and Clique. Our algorithms produce optimal solutions when provided with perfect predictions and their approximation ratios smoothly degrade with increasing prediction error. With small enough prediction error we achieve approximation guarantees that are beyond reach without predictions in the given time bounds, as exemplified by the NP-hardness and APX-hardness of many of the above problems. Although we show our approach to be optimal for this class of problems as a whole, there is a potential for exploiting specific structural properties of individual problems to obtain improved bounds; we demonstrate this on the Steiner Tree problem. We conclude with an empirical evaluation of our approach.

Paper Structure

This paper contains 41 sections, 14 theorems, 43 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Our results in more detail
  4. Approximation ratio.
  5. Prediction and prediction error.
  6. Minimization problems.
  7. Maximization problems.
  8. Lower bounds.
  9. Refined algorithm for Steiner Tree.
  10. Experimental results.
  11. Further remarks on our setting
  12. Learnability.
  13. Infeasible predictions.
  14. Related work
  15. Learning-augmented approximation algorithms.
  16. ...and 26 more sections

Key Result

Theorem 2

Let $\Pi$ be a minimization selection problem, and let $A$ be a $\rho$-approximation algorithm for $\Pi$ running in time $T(n)$. Then, there exists an $O(T(n))$-time learning-augmented approximation algorithm for $\Pi$ with the following guarantee: Upon receiving a (not necessarily feasible) predict

Figures (2)

  • Figure 1: Steiner tree problem with $k=n-1$ terminals (square) and one non-terminal vertex (circle). Outer edges have weight $2$, except one edge of weight $\beta$. Inner edges have weight $1+\epsilon$. Red edges are predicted. On the right are the respective Steiner trees output by Algorithm \ref{['alg:steiner']} with different parameters of $\alpha$.
  • Figure 2: Experimental evaluation of our refined Steiner Tree algorithm from Section \ref{['sec:steiner-tree']} (ALPS), with different values of the confidence parameter $\alpha$, compared to an equally fast classic approximation algorithm (Mehlhorn) and a much slower near-optimal solver (CIMAT). The x-axis represents the parameter $p$ controlling the prediction error: accurate predictions are to the left and erroneous predictions are to the right. The y-axis represents the value of the returned solution (the lower the better).

Theorems & Definitions (24)

  • Definition 1: Selection problem
  • Theorem 2
  • proof
  • Corollary 3
  • Theorem 4
  • proof
  • Corollary 5
  • Lemma 1
  • proof
  • Proposition 6: Mehlhorn88
  • ...and 14 more