Abstract
In this paper, we introduce the Maximum Matrix Contraction problem, where we aim to contract as much as possible a binary matrix in order to maximize its density. We study the complexity and the polynomial approximability of the problem. Especially, we prove this problem to be NP-Complete and that every algorithm solving this problem is at most a \(2\sqrt{n}\)-approximation algorithm where n is the number of ones in the matrix. We then focus on efficient algorithms to solve the problem: an integer linear program and three heuristics.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
The meaning of \([\![p; q ]\!]\) is the list \([p,p+1,\dots , q]\).
- 2.
The implementations can be found at https://github.com/mouton5000/MMCCode.
References
Pillai, A., Chick, J., Johanning, L., Khorasanchi, M., de Laleu, V.: Offshore wind farm electrical cable layout optimization. Eng. Optim. 47(12), 1689–1708 (2015)
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations, pp. 85–103. Springer, Heidelberg (1972)
Håstad, J.: Clique is hard to approximate within n\({\hat{}} (1-\epsilon \)). Acta Math. 182(1), 105–142 (1999)
Watel, D., Poirion, P.: The Maximum Matrix Contraction problem: Appendix. Technical report CEDRIC-16-3645, CEDRIC laboratory, CNAM, France (2016)
Lubin, M., Dunning, I.: Computing in operations research using Julia. INFORMS J. Comput. 27(2), 238–248 (2015)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Watel, D., Poirion, PL. (2016). The Maximum Matrix Contraction Problem. In: Cerulli, R., Fujishige, S., Mahjoub, A. (eds) Combinatorial Optimization. ISCO 2016. Lecture Notes in Computer Science(), vol 9849. Springer, Cham. https://doi.org/10.1007/978-3-319-45587-7_37
Download citation
DOI: https://doi.org/10.1007/978-3-319-45587-7_37
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-45586-0
Online ISBN: 978-3-319-45587-7
eBook Packages: Computer ScienceComputer Science (R0)