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Abstract. We study complexity of short sentences in Presburger arithmetic (Short-PA). Here by "short" we mean sentences with a bounded number of variables, quantifiers, inequalities and Boolean operations; the input consists only of the integers involved in the inequalities. We prove that assuming Kannan's partition can be found in polynomial time, the satisfiability of Short-PA sentences can be decided in polynomial time. Furthermore, under the same assumption, we show that the numbers of satisfying assignments of short Presburger sentences can also be computed in polynomial time.

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We study the computational complexity of short sentences in Presburger arithmetic (Short-PA). Here by "short" we mean sentences with a bounded number of variables, quantifiers, inequalities and Boolean operations; the input consists only of the integer coefficients involved in the linear inequalities. We prove that satisfiability of Short-PA sentences with m + 2 alternating quantifiers is Σ P m -complete or Π P m -complete, when the first quantifier is ∃ or ∀, respectively. Counting versions and restricted systems are also analyzed. Further application are given to hardness of two natural problems in Integer Optimization.