Combinatorial Optimization Problems with Balanced Regret

TL;DR

This paper introduces a generalized balanced regret framework for decision-making under uncertainty, extending min-max regret by comparing solutions affected by uncertainty, and analyzes its computational complexity and practical performance.

Contribution

It proposes a novel balanced regret approach for combinatorial problems, providing solution methods, complexity analysis, and empirical evaluation.

Findings

Balanced regret problems are NP-hard for selection problems.

Balanced regret solutions offer a beneficial trade-off in performance measures.

The approach broadens the applicability of robust decision-making under uncertainty.

Abstract

For decision making under uncertainty, min-max regret has been established as a popular methodology to find robust solutions. In this approach, we compare the performance of our solution against the best possible performance had we known the true scenario in advance. We introduce a generalization of this setting which allows us to compare against solutions that are also affected by uncertainty, which we call balanced regret. Using budgeted uncertainty sets, this allows for a wider range of possible alternatives the decision maker may choose from. We analyze this approach for general combinatorial problems, providing an iterative solution method and insights into solution properties. We then consider a type of selection problem in more detail and show that, while the classic regret setting with budgeted uncertainty sets can be solved in polynomial time, the balanced regret problem becomes NP-hard. In computational experiments using random and real-world data, we show that balanced regret solutions provide a useful trade-off for the performance in classic performance measures.