On Scenario Aggregation to Approximate Robust Optimization Problems

TL;DR

This paper introduces an enhanced scenario aggregation technique that improves approximation bounds for robust combinatorial optimization problems, extending the midpoint method to achieve arbitrarily close to optimal solutions.

Contribution

It presents a simple extension of the midpoint method using scenario aggregation, achieving better approximation bounds for robust min-max and min-max regret problems.

Findings

Improves approximation bounds from K to εK for any ε > 0.

Applicable to both min-max and min-max regret problems.

Enhances robustness in combinatorial optimization solutions.

Abstract

As most robust combinatorial min-max and min-max regret problems with discrete uncertainty sets are NP-hard, research into approximation algorithm and approximability bounds has been a fruitful area of recent work. A simple and well-known approximation algorithm is the midpoint method, where one takes the average over all scenarios, and solves a problem of nominal type. Despite its simplicity, this method still gives the best-known bound on a wide range of problems, such as robust shortest path, or robust assignment problems. In this paper we present a simple extension of the midpoint method based on scenario aggregation, which improves the current best $K$-approximation result to an $(\varepsilon K)$-approximation for any desired $\varepsilon > 0$. Our method can be applied to min-max as well as min-max regret problems.