Randomized Minmax Regret for Combinatorial Optimization Under Uncertainty

Abstract

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.

A double oracle approach to minmax regret optimization problems with interval data

“…Note that Mastin et al investigated a similar game-theoretic view of robust optimization [9]. They showed how to compute a randomized Nash equilibrium by using a mathematical programming formulation involving an exponential number of constraints.…”

“…Nevertheless, for the game considered here between the x-player and the c-player, the complete linear program, that involves as many variables as there are pure strategies for both players, would be huge, and it is therefore not worth considering its generation in extension. To tackle this issue, Mastin et al [9] relies on a cutting-plane method.…”

“…This separation oracle is also polynomial time if one can optimize over X in polynomial time. As noted by Mastin et al [9], it implies that the complexity of determining a mixed NE is polynomial if the standard version of the tackled problem is polynomially solvable.…”

“…Our approach relies on a game-theoretic view of robust optimization. Such a gametheoretic view has already been adopted once in the literature by Mastin et al [9]. They view robust optimization as a two-player zero-sum game where one player (optimizing player) selects a feasible solution, and the other one (adversary) selects a possible scenario for the values of the parameters.…”

“…Section 2 recalls the definition of a minmax regret problem and introduces the notations. In section 3, a game theoretic view of minmax regret optimization, similar to the one proposed by Mastin et al [9], is developed. This view helps us derive a precise lower bound on the objective value of a minmax regret solution.…”