On the Complexity of Quantified Integer Programming

Authors Dmitry Chistikov, Christoph Haase

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Dmitry Chistikov
Christoph Haase

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Dmitry Chistikov and Christoph Haase. On the Complexity of Quantified Integer Programming. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 94:1-94:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


Quantified integer programming is the problem of deciding assertions of the form Q_k x_k ... forall x_2 exists x_1 : A * x >= c where vectors of variables x_k,..,x_1 form the vector x, all variables are interpreted over N (alternatively, over Z), and A and c are a matrix and vector over Z of appropriate sizes. We show in this paper that quantified integer programming with alternation depth k is complete for the kth level of the polynomial hierarchy.
  • integer programming
  • semi-linear sets
  • Presburger arithmetic
  • quantifier elimination


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