Abstract
Quantified linear programs (QLPs) are linear programs with variables being either existentially or universally quantified. The integer variant is PSPACE-complete, and the problem is similar to games like chess, where an existential and a universal player have to play a two-person-zero-sum game. At the same time, a QLP with n variables is a variant of a linear program living in \({\hbox{\rm I\kern - 0.15em R}}^n\), and it has strong similarities with multistage-stochastic programs with variable right-hand side. We show for the continuous case that the union of all winning policies of the existential player forms a polytope in \({\hbox{\rm I\kern - 0.15em R}}^n\), that its vertices are games of so called extremal strategies, and that these vertices can be encoded with polynomially many bits. The latter allows the conclusion that solving a QLP is in PSPACE. The hardness of the problem stays unknown.
Research partially supported by German Research Foundation (DFG) funded SFB 805.
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Lorenz, U., Martin, A., Wolf, J. (2010). Polyhedral and Algorithmic Properties of Quantified Linear Programs. In: de Berg, M., Meyer, U. (eds) Algorithms – ESA 2010. ESA 2010. Lecture Notes in Computer Science, vol 6346. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15775-2_44
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