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Polynomially solvable cases of the constant rank unconstrained quadratic 0-1 programming problem

Author

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  • Eranda Çela

    (Technische Universität Graz)

  • Bettina Klinz

    (Technische Universität Graz)

  • Christophe Meyer

    (Université de Montréal)

Abstract

In this paper we consider the constant rank unconstrained quadratic 0-1 optimization problem, CR-QP01 for short. This problem consists in minimizing the quadratic function 〈x, Ax〉 + 〈c, x〉 over the set {0,1} n where c is a vector in ℝ n and A is a symmetric real n × n matrix of constant rank r. We first present a pseudo-polynomial algorithm for solving the problem CR-QP01, which is known to be NP-hard already for r = 1. We then derive two new classes of special cases of the CR-QP01 which can be solved in polynomial time. These classes result from further restrictions on the matrix A. Finally we compare our algorithm with the algorithm of Allemand et al. (2001) for the CR-QP01 with negative semidefinite A and extend the range of applicability of the latter algorithm. It turns out that neither of the two algorithms dominates the other with respect to the class of instances which can be solved in polynomial time.

Suggested Citation

  • Eranda Çela & Bettina Klinz & Christophe Meyer, 2006. "Polynomially solvable cases of the constant rank unconstrained quadratic 0-1 programming problem," Journal of Combinatorial Optimization, Springer, vol. 12(3), pages 187-215, November.
    • Handle: RePEc:spr:jcomop:v:12:y:2006:i:3:d:10.1007_s10878-006-9625-0
    DOI: 10.1007/s10878-006-9625-0
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    References listed on IDEAS

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    1. Ferrez, J.-A. & Fukuda, K. & Liebling, Th.M., 2005. "Solving the fixed rank convex quadratic maximization in binary variables by a parallel zonotope construction algorithm," European Journal of Operational Research, Elsevier, vol. 166(1), pages 35-50, October.
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    Cited by:

    1. X. Sun & C. Liu & D. Li & J. Gao, 2012. "On duality gap in binary quadratic programming," Journal of Global Optimization, Springer, vol. 53(2), pages 255-269, June.
    2. Ben-Ameur, Walid & Neto, José, 2010. "Spectral bounds for unconstrained (-1,1)-quadratic optimization problems," European Journal of Operational Research, Elsevier, vol. 207(1), pages 15-24, November.

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