Abstract
A combinatorial optimization problem, called the Bandpass Problem, is introduced. Given a rectangular matrix A of binary elements {0,1} and a positive integer B called the Bandpass Number, a set of B consecutive non-zero elements in any column is called a Bandpass. No two bandpasses in the same column can have common rows.
The Bandpass problem consists of finding an optimal permutation of rows of the matrix, which produces the maximum total number of bandpasses having the same given bandpass number in all columns.
This combinatorial problem arises in considering the optimal packing of information flows on different wavelengths into groups to obtain the highest available cost reduction in design and operating the optical communication networks using wavelength division multiplexing technology. Integer programming models of two versions of the bandpass problems are developed. For a matrix A with three or more columns the Bandpass problem is proved to be NP-hard. For matrices with two or one column a polynomial algorithm solving the problem to optimality is presented. For the general case fast performing heuristic polynomial algorithms are presented, which provide near optimal solutions, acceptable for applications. High quality of the generated heuristic solutions has been confirmed in the extensive computational experiments.
As an NP-hard combinatorial optimization problem with important applications the Bandpass problem offers a challenge for researchers to develop efficient computational solution methods. To encourage the further research a Library of Bandpass Problems has been developed. The Library is open to public and consists of 90 problems of different sizes (numbers of rows, columns and density of non-zero elements of matrix A and bandpass number B), half of them with known optimal solutions and the second half, without.
Similar content being viewed by others
References
Babayev DA, Bell GI, Nuriyev UG, Kurt M (2007) Library of bandpass problems. http://sci.ege.edu.tr/~math/BandpassProblemsLibrary/
Bell G, Babayev DA (2004) Bandpass problem. In: Annual INFORMS meeting, October 2004, Denver, CO, USA
Cheung NK, Nosu G, Winzer G (eds) (1990) In: IEEE JSAC: special issue on dense WDM networks, vol 8, August 1990
Cook SA (eds) (1971) The complexity of theorem-proving procedures. In: Annual ACM symposium on theory of computing, pp 151–158
Czyzyk J, Mesnier M, Moré J (1998) The NEOS server. IEEE J Comput Sci Eng 5:68–75
Dolan E (2001) The NEOS server 4.0 administrative guide. Technical Memorandum ANL/MCS-TM-250, Mathematics and Computer Science Division, Argonne National Laboratory, May 2001
Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. Freeman, New York
Goralski WJ (1997) SONET, a guide to synchronous optical networks. Series on computer communications. McGrow-Hill, New York, 484 pp
Gropp W, Moré J (1997) Optimization environments and the NEOS server. In: Buhmann MD, Iserles A (eds) Approximation theory and optimization. Cambridge University Press, Cambridge, pp 167–182
Kaminov IP et al. (1996) A wideband all-optical WDM network. IEEE JSAC/JLT 14(5):780–799. Special issue on optical networks
Knuth DE (1973) The art of computer programming, volume 1/fundamental algorithms. Addison-Wesley, Reading, 636 pp
Nemhauser GL, Woolsey LA (ed) (1988) Integer and combinatorial optimization. Wiley, New York, 763 pp
Nosu KL, O’Mahony MJ (eds) (1994) In: IEEE Communications Magazine: special issue on optically multiplexed networks, vol 32, Dec 1994
Papadimitriou CH, Steiglitz K (1998) Combinatorial optimization: algorithms and complexity. Dover, New York, 512 pp
Ramaswami R, Sivarajan KI (1998) Optical networks. A practical perspective. Morgan Kaufman, San Francisco, 632 pp
Woolsey LA (1998) Integer programming. Wiley, New York, 264 pp
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Babayev, D.A., Bell, G.I. & Nuriyev, U.G. The bandpass problem: combinatorial optimization and library of problems. J Comb Optim 18, 151–172 (2009). https://doi.org/10.1007/s10878-008-9143-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-008-9143-3