article

The boolean quadric polytope: Some characteristics, facets and relatives

Author:

Manfred PadbergAuthors Info & Claims

Mathematical Programming: Series A and B, Volume 45, Issue 1-3

Pages 139 - 172

Published: 01 August 1989 Publication History

Abstract

We study unconstrained quadratic zero---one programming problems havingn variables from a polyhedral point of view by considering the Boolean quadric polytope QPn inn(n+1)/2 dimensions that results from the linearization of the quadratic form. We show that QPn has a diameter of one, descriptively identify three families of facets of QPn and show that QPn is symmetric in the sense that all facets of QPn can be obtained from those that contain the origin by way of a mapping. The naive linear programming relaxation QPnLP of QPn is shown to possess the Trubin-property and its extreme points are shown to be {0,1/2,1}-valued. Furthermore, O(n3) facet-defining inequalities of QPn suffice to cut off all fractional vertices of QPnLP, whereas the family of facets described by us has at least O(3n) members. The problem is also studied for sparse quadratic forms (i.e. when several or many coefficients are zero) and conditions are given for the previous results to carry over to this case. Polynomially solvable problem instances are discussed and it is shown that the known polynomiality result for the maximization of nonnegative quadratic forms can be obtained by simply rounding the solution to the linear programming relaxation. In the case that the graph induced by the nonzero coefficients of the quadratic form is series-parallel, a complete linear description of the associated Boolean quadric polytope is given. The relationship of the Boolean quadric polytope associated to sparse quadratic forms with the vertex-packing polytope is discussed as well.

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Index Terms

The boolean quadric polytope: Some characteristics, facets and relatives
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
  2. Mathematical analysis
    1. Mathematical optimization
      1. Continuous optimization
        Linear programming
2. Theory of computation
  1. Design and analysis of algorithms
    1. Mathematical optimization
  2. Randomness, geometry and discrete structures

Index terms have been assigned to the content through auto-classification.

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Published In

cover image Mathematical Programming: Series A and B

Mathematical Programming: Series A and B Volume 45, Issue 1-3

August 1989

556 pages

ISSN:0025-5610

Issue’s Table of Contents

Copyright © Copyright © 1989 North-Holland.

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 August 1989

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Meshi OLondon BWeller ASontag D(2019)Train and test tightness of LP relaxations in structured predictionThe Journal of Machine Learning Research10.5555/3322706.332271920:1(480-513)Online publication date: 1-Jan-2019
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Lee JSkipper DSpeakman E(2018)Algorithmic and modeling insights via volumetric comparison of polyhedral relaxationsMathematical Programming: Series A and B10.5555/3238577.3238588170:1(121-140)Online publication date: 1-Jul-2018
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Anstreicher K(2018)Maximum-entropy sampling and the Boolean quadric polytopeJournal of Global Optimization10.1007/s10898-018-0662-x72:4(603-618)Online publication date: 1-Dec-2018
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