A note on the complexity of two-stage stochastic linear optimization with small second stage

Abstract: Two-stage stochastic linear optimization is known to be #P-hard when all involved random variables are independently and uniformly distributed over intervals, even with fixed recourse. We show that this problem is actually #P-hard in the strong sense. More surprisingly, this hardness persists when the random vector is one-dimensional, i.e., uniformly distributed over a single interval. To obtain this result, we show that computing the area of a two-dimensional polytope given by a compact extended formulation is strongly #P-hard. Furthermore, we obtain the same complexity result in case the number of second-stage constraints is fixed (for a problem in standard form), while fixing the number of second-stage variables leads to a weakly #P-hard problem. Finally, if both the dimension of the random vector and the number of second-stage constraints are fixed, the problem turns out to be tractable.