A Novel Application for Game Tree Search - Exploiting Pruning Mechanisms for Quantified Integer Programs

Abstract

We investigate pruning in search trees of so-called quantified integer linear programs (QIPs). QIPs consist of a set of linear inequalities and a minimax objective function, where some variables are existentially and others are universally quantified. They can be interpreted as two-person zero-sum games between an existential and a universal player on the one hand, or multistage optimization problems under uncertainty on the other hand. Solutions are so-called winning strategies for the existential player that specify how to react on moves of the universal player - i.e. certain assignments of universally quantified variables - to certainly win the game. QIPs can be solved with the help of game tree search that is enhanced with non-chronological back-jumping. We develop and theoretically substantiate pruning techniques based upon (algebraic) properties similar to pruning mechanisms known from linear programming and quantified boolean formulas. The presented Strategic Copy-Pruning mechanism allows to \textit{implicitly} deduce the existence of a strategy in linear time (by static examination of the QIP-matrix) without explicitly traversing the strategy itself. We show that the implementation of our findings can massively speed up the search process.