Abstract
Integer programs (IPs) are one of the fundamental tools used to solve combinatorial problems in theory and practice. Understanding the structure of solutions of IPs is thus helpful to argue about the existence of solutions with a certain simple structure, leading to significant algorithmic improvements. Typical examples for such structural properties are solutions that use a specific type of variable very often or solutions that only contain few non-zero variables. The last decade has shown the usefulness of this method. In this paper we summarize recent progress for structural properties and their algorithmic implications in the area of approximation algorithms and fixed parameter tractability. Concretely, we show how these structural properties lead to optimal approximation algorithms for the classical Makespan Scheduling scheduling problem and to exact polynomial-time algorithm for the Bin Packing problem with a constant number of different item sizes.
This work was partially supported by the Swiss National Science Foundation (SNSF) within the project Convexity, geometry of numbers, and the complexity of integer programming (Nr. 163071) and DFG Project “Entwicklung und Analyse von effizienten polynomiellen Approximationsschemata für Scheduling- und verwandte Optimierungsprobleme”, Ja 612/14-2.
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
References
Aliev, I., De Loera, J., Eisenbrand, F., Oertel, T., Weismantel, R.: The support of integer optimal solutions. arxiv:1712.08923 (2017)
Chen, L., Jansen, K., Zhang, G.: On the optimality of approximation schemes for the classical scheduling problem. In: Proceedings of SODA. SIAM, pp. 657–668 (2014)
Eisenbrand, F., Shmonin, G.: Carathodory bounds for integer cones. Oper. Res. Lett. 34(5), 564–568 (2006)
Filippi, C.: On the bin packing problem with a fixed number of object weights. Eur. J. Oper. Res. (EJOR) 181(1), 117–126 (2007)
Goemans, M.X., Rothvoß, T.: Polynomiality for bin packing with a constant number of item types. In: Proceedings of SODA, pp. 830–839 (2014)
Jansen, K., Klein, K.: About the structure of the integer cone and its application to bin packing. In: Proceedings of SODA. SIAM, pp. 1571–1581 (2017)
Jansen, K., Klein, K., Verschae, J.: Closing the gap for makespan scheduling via sparsification techniques. In: Proceedings of ICALP, LIPIcs, vol. 55, pp. 72:1–72:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2016)
Kannan, R.: Minkowskis convex body theorem and integer programming. Math. Oper. Res. 12(3), 415–440 (1987)
Lenstra, H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8, 538–548 (1983)
McCormick, S.T., Smallwood, S.R., Spieksma, F.C.R.: Polynomial algorithms for multiprocessor scheduling with a small number of job lengths. In: Proceedings of SODA. SIAM, pp. 509–517 (1997)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Berndt, S., Klein, KM. (2018). Using Structural Properties for Integer Programs. In: Manea, F., Miller, R., Nowotka, D. (eds) Sailing Routes in the World of Computation. CiE 2018. Lecture Notes in Computer Science(), vol 10936. Springer, Cham. https://doi.org/10.1007/978-3-319-94418-0_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-94418-0_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-94417-3
Online ISBN: 978-3-319-94418-0
eBook Packages: Computer ScienceComputer Science (R0)