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On the computational complexities of Quantified Integer Programming variants

In this paper, we study Quantified Integer Programming (QIP) and Quantified Integer Implication (QII) from the perspective of computational complexity. In particular, we show that the restricted language of QIIs is expressive enough to fully capture Presburger Arithmetic. Secondly, we establish the computational complexity of QIP over polytopes, i.e., cases where an upper and lower bound can always be derived for existential variables. Thirdly, we investigate the complexities of partially bounded and unbounded variants of QIPs and QIIs. Finally, we examine the connections between QIPs and QIIs with limited quantifier alternations and the polynomial hierarchy.