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DOI: 10.4230/LIPIcs.ICALP.2024.132

Integer Linear-Exponential Programming in NP by Quantifier Elimination

Authors: Dmitry Chistikov, Alessio Mansutti, and Mikhail R. Starchak

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract

This paper provides an NP procedure that decides whether a linear-exponential system of constraints has an integer solution. Linear-exponential systems extend standard integer linear programs with exponential terms 2^x and remainder terms (x mod 2^y). Our result implies that the existential theory of the structure (ℕ,0,1,+,2^(⋅),V_2(⋅,⋅), ≤) has an NP-complete satisfiability problem, thus improving upon a recent EXPSPACE upper bound. This theory extends the existential fragment of Presburger arithmetic with the exponentiation function x ↦ 2^x and the binary predicate V_2(x,y) that is true whenever y ≥ 1 is the largest power of 2 dividing x. Our procedure for solving linear-exponential systems uses the method of quantifier elimination. As a by-product, we modify the classical Gaussian variable elimination into a non-deterministic polynomial-time procedure for integer linear programming (or: existential Presburger arithmetic).

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Dmitry Chistikov, Alessio Mansutti, and Mikhail R. Starchak. Integer Linear-Exponential Programming in NP by Quantifier Elimination. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 132:1-132:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chistikov_et_al:LIPIcs.ICALP.2024.132,
  author =	{Chistikov, Dmitry and Mansutti, Alessio and Starchak, Mikhail R.},
  title =	{{Integer Linear-Exponential Programming in NP by Quantifier Elimination}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{132:1--132:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.132},
  URN =		{urn:nbn:de:0030-drops-202758},
  doi =		{10.4230/LIPIcs.ICALP.2024.132},
  annote =	{Keywords: decision procedures, integer programming, quantifier elimination}
}

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Documents authored by Starchak, Mikhail R.

Integer Linear-Exponential Programming in NP by Quantifier Elimination

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