Analyzing Selected Quantified Integer Programs

In this paper, we introduce a problem called Quantified Integer Programming, which generalizes the Quantified Satisfiability problem (QSAT). In a Quantified Integer Program (QIP), the program variables can assume arbitrary integral values, as opposed to the boolean values that are assumed by the variables of an instance of QSAT. QIPs naturally represent 2-person integer matrix games. The Quantified Integer Programming problem is PSPACE-hard in general, since the QSAT problem is PSPACE-complete. We focus on analyzing various special cases of the general problem, with a view to discovering subclasses that are tractable. Subclasses of the general QIP problem are obtained by restricting either the constraint matrix or the quantifier specification. We show that if the constraint matrix is totally unimodular, the problem of deciding a QIP can be solved in polynomial time.

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