Approximation Algorithms for Min-max-min Robust Optimization and K-Adaptability under Objective Uncertainty

TL;DR

This paper develops approximation algorithms for min-max-min and k-adaptability robust optimization problems with uncertain costs, providing theoretical guarantees and practical efficiency for problems with intermediate k values.

Contribution

It introduces approximation algorithms with problem-specific guarantees for intermediate k, extending understanding of robust optimization complexity and solution methods.

Findings

Algorithms achieve small optimality gaps in seconds

Bounds show polynomial-time solvability under certain conditions

Experimental results validate efficiency and effectiveness

Abstract

In this work we investigate the min-max-min robust optimization problem and the k-adaptability robust optimization problem for binary problems with uncertain costs. The idea of the first approach is to calculate a set of k feasible solutions which are worst-case optimal if in each possible scenario the best of the k solutions is implemented. It is known that the min-max-min robust problem can be solved efficiently if k is at least the dimension of the problem, while it is theoretically and computationally hard if k is small. However, nothing is known about the intermediate case, i.e. k lies between one and the dimension of the problem. We approach this open question and present an approximation algorithm which achieves good problem-specific approximation guarantees for the cases where k is close to or where k is a fraction of the dimension. The derived bounds can be used to show that the min-max-min robust problem is solvable in oracle-polynomial time under certain conditions even if k is smaller than the dimension. We extend the previous results to the robust k-adaptability problem. As a consequence we can provide bounds on the number of necessary second-stage policies to approximate the exact two-stage robust problem. We derive an approximation algorithm for the k-adaptability problem which has similar guarantees as for the min-max-min problem. Finally, we test both algorithms on knapsack and shortest path problems and related two-stage variants. The experiments show that both algorithms calculate solutions with relatively small optimality gap in seconds.