# Randomized Minmax Regret for Combinatorial Optimization Under Uncertainty

arXiv: https://arxiv.org/abs/1401.7043 · HTML: https://mlsanity.com/paper/1401.7043/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jan 2014.

## Built on by (most cited first, 1 of 1)

- [Robust Optimization with Diffusion Models for Green Security](https://mlsanity.com/paper/2503.05730/) (2025)

## Abstract (arXiv)

The minmax regret problem for combinatorial optimization under uncertainty can be viewed as a zero-sum game played between an optimizing player and an adversary, where the optimizing player selects a solution and the adversary selects costs with the intention of maximizing the regret of the player. The existing minmax regret model considers only deterministic solutions/strategies, and minmax regret versions of most polynomial solvable problems are NP-hard. In this paper, we consider a randomized model where the optimizing player selects a probability distribution (corresponding to a mixed strategy) over solutions and the adversary selects costs with knowledge of the player's distribution, but not its realization. We show that under this randomized model, the minmax regret version of any polynomial solvable combinatorial problem becomes polynomial solvable. This holds true for both the interval and discrete scenario representations of uncertainty. Using the randomized model, we show new proofs of existing approximation algorithms for the deterministic model based on primal-dual approaches. Finally, we prove that minmax regret problems are NP-hard under general convex uncertainty.