Robust Combinatorial Optimization Problems Under Budgeted Interdiction Uncertainty

TL;DR

This paper investigates robust combinatorial optimization under budgeted interdiction uncertainty, revealing that certain problem variants are NP-hard despite efficient adversarial problem solutions, with some models scaling well in practice.

Contribution

It introduces a novel variant of discrete budgeted uncertainty with weighted constraints, analyzing its computational complexity and practical scalability.

Findings

Adversarial problems are solvable in linear time.

Robust problems are NP-hard and not approximable.

Some models scale well despite theoretical hardness.

Abstract

In robust combinatorial optimization, we would like to find a solution that performs well under all realizations of an uncertainty set of possible parameter values. How we model this uncertainty set has a decisive influence on the complexity of the corresponding robust problem. For this reason, budgeted uncertainty sets are often studied, as they enable us to decompose the robust problem into easier subproblems. We propose a variant of discrete budgeted uncertainty for cardinality-based constraints or objectives, where a weight vector is applied to the budget constraint. We show that while the adversarial problem can be solved in linear time, the robust problem becomes NP-hard and not approximable. We discuss different possibilities to model the robust problem and show experimentally that despite the hardness result, some models scale relatively well in the problem size.