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On Vertices of the Simple Boolean Quadric Polytope Extension

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Optimization Problems and Their Applications (OPTA 2018)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 871))

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Abstract

Following the seminal work of Padberg on the Boolean quadric polytope BQP and its LP relaxation \(BQP_{LP}\), we consider a natural extension: the polytopes SATP and \(SATP_{LP}\), with \(BQP_{LP}\) being a projection of \(SATP_{LP}\) face (and BQP – projection of SATP face). Various special instances of 3-SAT problem like NAE-3-SAT, 1-in-3-SAT, weighted MAX-3-SAT, and others can be solved by integer programming over \(SATP_{LP}\). We consider the properties of SATP 1-skeleton and \(SATP_{LP}\) fractional vertices. Like \(BQP_{LP}\), the polytope \(SATP_{LP}\) has the Trubin-property being quasi-integral (1-skeleton of SATP is a subset of 1-skeleton of \(SATP_{LP}\)). However, unlike BQP, not all vertices of SATP are pairwise adjacent, the diameter of SATP equals 2, and the clique number of 1-skeleton is superpolynomial in dimension. It is known that the fractional vertices of \(BQP_{LP}\) are half-integral (0, 1 or 1/2 valued). We establish that the denominators of \(SATP_{LP}\) fractional vertices can take any integer values.

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Acknowledgments

The research is supported by the grant of the President of the Russian Federation MK-2620.2018.1.

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Authors and Affiliations

  1. Department of Discrete Analysis, P.G. Demidov Yaroslavl State University, Yaroslavl, Russia

    Andrei V. Nikolaev

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  1. Andrei V. Nikolaev
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Correspondence to Andrei V. Nikolaev .

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Editors and Affiliations

  1. Sobolev Institute of Mathematics SB RAS, Omsk, Russia

    Anton Eremeev

  2. Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    Michael Khachay

  3. Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia

    Yury Kochetov

  4. Industrial and Systems Engineering, University of Florida, Gainesville, Florida, USA

    Panos Pardalos

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Nikolaev, A.V. (2018). On Vertices of the Simple Boolean Quadric Polytope Extension. In: Eremeev, A., Khachay, M., Kochetov, Y., Pardalos, P. (eds) Optimization Problems and Their Applications. OPTA 2018. Communications in Computer and Information Science, vol 871. Springer, Cham. https://doi.org/10.1007/978-3-319-93800-4_13

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  • DOI: https://doi.org/10.1007/978-3-319-93800-4_13

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-93799-1

  • Online ISBN: 978-3-319-93800-4

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