Complexity of the min-max (regret) versions of cut problems

This paper investigates the complexity of the min-max and min-max regret versions of the s-t min cut, and min Cut problems. Even if the underlying problems are closely related and both polynomial, we show that the complexity of their min-max and min-max regret versions, for a constant number of scenarios, are quite contrasted since they are respectively strongly NP-hard and polynomial. Thus, we exhibit the first polynomial problem, s-t min cut, whose min-max (regret) versions are strongly NP-hard. Also, min cut is one of the few polynomial problems whose min-max (regret) versions remain polynomial. However, these versions become strongly NP-hard for a non constant number of scenarios. In the interval data case, min-max versions are trivially polynomial. Moreover, for min-max regret versions, we obtain the same contrasted result as for a constant number of scenarios: min-max regret s-t cut is strongly NP-hard whereas min-max regret cut is polynomial.