A lower bound for a constrained quadratic 0–1 minimization problem

Given a quadratic pseudo-Boolean function f( x 1, …, x n ) written as a multilinear polynomial in its variables, Hammer et al. [7]have studied, in their paper “Roof duality, complementation and persistency in quadratic 0–1 optimization”, the greatest constant c such that there exists a quadratic pos...

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Bibliographic Details
Published in:Discrete Applied Mathematics Vol. 74; no. 2; pp. 135 - 146
Main Authors: Billionnet, Alain, Faye, Alain
Format: Journal Article
Language:English
Published: Elsevier B.V 18.04.1997
Elsevier
Subjects:
Computational Complexity
Computer Science
Constrained zero-one quadratic programming
Linear programming
Lower bound
Roof duality
Lower bound
Linear programming
Constrained zero-one quadratic programming
Roof duality
ISSN:0166-218X, 1872-6771
Online Access:Get full text
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Summary:Given a quadratic pseudo-Boolean function f( x 1, …, x n ) written as a multilinear polynomial in its variables, Hammer et al. [7]have studied, in their paper “Roof duality, complementation and persistency in quadratic 0–1 optimization”, the greatest constant c such that there exists a quadratic posiform φ satisfying f = c + φ for all xϵ {0, 1} n . Obviously c is a lower bound to the minimum of f. In this paper we consider the problem of minimizing a quadratic pseudo- Boolean function subject to the cardinality constraint ∑ i = 1, n x i = k and we propose a linear programming method to compute the greatest constant c such that there exists a quadratic posiform φ satisfying f = c + φ for all x ϵ {0, 1} n with ∑ i = 1, n x i = k. As in the unconstrained case c is a lower bound to the optimum. Some computational tests showing how sharp this bound is in practice are reported.
ISSN:0166-218X
1872-6771
DOI:10.1016/S0166-218X(96)00026-1