Abstract
A critical step in a cutting plane algorithm is separation, i.e., establishing whether a given vector x violates an inequality belonging to a specific class. It is customary to express the time complexity of a separation algorithm in the number of variables n. Here, we argue that a separation algorithm may instead process the vector containing the positive components of x, denoted as supp(x), which offers a more compact representation, especially if x is sparse; we also propose to express the time complexity in terms of |supp(x)|. Although several well-known separation algorithms exploit the sparsity of x, we revisit this idea in order to take sparsity explicitly into account in the time-complexity of separation and also design faster algorithms. We apply this approach to two classes of facet-defining inequalities for the three-index assignment problem, and obtain separation algorithms whose time complexity is linear in |supp(x)| instead of n. We indicate that this can be generalized to the axial k-index assignment problem and we show empirically how the separation algorithms exploiting sparsity improve on existing ones by running them on the largest instances reported in the literature.
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Acknowledgments
We are thankful to Armin Fügenschuh for providing the routines to generate instances used in [15], and to Yves Crama for stimulating discussions on this subject.
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This paper is an improved version of an extended abstract that appeared as “Fast separation algorithms for three index assignment problems” in the proceedings of ISCO 2012, LNCS 7422, pp. 189–200, 2012.
This research has been co-financed by the European Union (European Social Fund ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)—Research Funding Program: Thales: Investing in knowledge society through the European Social Fund (MIS: 380232), and this research has been supported by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy.
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Dokka, T., Mourtos, I. & Spieksma, F.C.R. Fast separation for the three-index assignment problem. Math. Prog. Comp. 9, 39–59 (2017). https://doi.org/10.1007/s12532-016-0106-x
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DOI: https://doi.org/10.1007/s12532-016-0106-x