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The boolean quadric polytope: Some characteristics, facets and relatives

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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 QPn LP of QPn is shown to possess the Trubin-property and its extreme points are shown to be {0,1/2,1}-valued. Furthermore, O(n 3) facet-defining inequalities of QPn suffice to cut off all fractional vertices of QPn LP, 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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Authors and Affiliations

  1. Stern School of Business, New York University, Washington Square, 10003, New York, NY, USA

    Manfred Padberg

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  1. Manfred Padberg
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Additional information

Dedicato ad Alan Hoffman in ammirazione per l'uomo e le sue opere.

Partial support under NSF Grant Nos. DMS-8508955 and ECS-8615438 and ONR Grant No. R&T-41663.

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Padberg, M. The boolean quadric polytope: Some characteristics, facets and relatives. Mathematical Programming 45, 139–172 (1989). https://doi.org/10.1007/BF01589101

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