  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.
