Solution Techniques for Quantified Linear Programs and the Links to Gaming

Abstract

Quantified linear programs (QLPs) are linear programs (LPs) with variables being either existentially or universally quantified. QLPs are two-person zero-sum games between an existential and a universal player on the one side, and convex multistage decision problems on the other side. Solutions of feasible QLPs are so called winning strategies for the existential player that specify how to react on moves – well-thought fixations of universally quantified variables – of the universal player to be sure to win the game. To find a certain best strategy among different winning strategies, we propose the extension of the QLP decision problem by an objective function. To solve the resulting QLP optimization problem, we exploit the problem’s hybrid nature and combine linear programming techniques with solution techniques from game-tree search. As a result, we present an extension of the Nested Benders Decomposition algorithm by the \(\alpha \beta \)-algorithm and its heuristical move-ordering as used in game-tree search to solve minimax trees. The applicability of our method to both QLPs and models of PSPACE-complete games such as Connect6 is examined in an experimental evaluation.