Experimental Analysis of LP Scaling Methods Based on Circuit Imbalance Minimization

Authors Jakub Komárek , Martin Koutecký

Thumbnail PDF


  • Filesize: 1.01 MB
  • 21 pages

Document Identifiers

Author Details

Jakub Komárek
  • Computer Science Institute, Charles University, Prague, Czech Republic
Martin Koutecký
  • Computer Science Institute, Charles University, Prague, Czech Republic

Cite AsGet BibTex

Jakub Komárek and Martin Koutecký. Experimental Analysis of LP Scaling Methods Based on Circuit Imbalance Minimization. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 18:1-18:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


Linear programming (LP) is a fundamental problem with rich theory and wide applications. A ubiquitous technique in LP is scaling, where the input instance is transformed in some way to make its solution easier. Dadush et al. [STOC '20] have recently devised an algorithm which scales the columns of the constraint matrix of a linear program in a way that aims to minimize the circuit imbalance measure, a matrix condition number of growing theoretical interest. They show that this rescaling achieves favorable theoretical guarantees for certain LP algorithms. We follow up on their work in an experimental manner. First, we have implemented their algorithm, overcoming several engineering obstacles. Next, we have used our implementation to obtain a rescaling of 142 publicly available instances. Finally, we have performed experiments evaluating the effects of the obtained rescalings on the runtime of real-world LP solvers, and we have evaluated their quality with regard to the circuit imbalance measure.

Subject Classification

ACM Subject Classification
  • Theory of computation → Linear programming
  • Linear programming
  • scaling
  • circuit imbalance measure


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Ksenia Bestuzheva, Mathieu Besançon, Wei-Kun Chen, Antonia Chmiela, Tim Donkiewicz, Jasper van Doornmalen, Leon Eifler, Oliver Gaul, Gerald Gamrath, Ambros Gleixner, Leona Gottwald, Christoph Graczyk, Katrin Halbig, Alexander Hoen, Christopher Hojny, Rolf van der Hulst, Thorsten Koch, Marco Lübbecke, Stephen J. Maher, Frederic Matter, Erik Mühmer, Benjamin Müller, Marc E. Pfetsch, Daniel Rehfeldt, Steffan Schlein, Franziska Schlösser, Felipe Serrano, Yuji Shinano, Boro Sofranac, Mark Turner, Stefan Vigerske, Fabian Wegscheider, Philipp Wellner, Dieter Weninger, and Jakob Witzig. The SCIP Optimization Suite 8.0. Technical report, Optimization Online, December 2021. URL: http://www.optimization-online.org/DB_HTML/2021/12/8728.html.
  2. Steffen Borgwardt, Cornelius Brand, Andreas Emil Feldmann, and Martin Koutecký. A note on the approximability of deepest-descent circuit steps. Oper. Res. Lett., 49(3):310-315, 2021. URL: https://doi.org/10.1016/J.ORL.2021.02.003.
  3. Steffen Borgwardt, Jesús A. De Loera, Elisabeth Finhold, and Jacob Miller. The hierarchy of circuit diameters and transportation polytopes. Discret. Appl. Math., 240:8-24, 2018. URL: https://doi.org/10.1016/J.DAM.2015.10.017.
  4. Steffen Borgwardt, Tamon Stephen, and Timothy Yusun. On the circuit diameter conjecture. Discret. Comput. Geom., 60(3):558-587, 2018. URL: https://doi.org/10.1007/S00454-018-9995-Y.
  5. Steffen Borgwardt and Charles Viss. A polyhedral model for enumeration and optimization over the set of circuits. Discret. Appl. Math., 308:68-83, 2022. URL: https://doi.org/10.1016/J.DAM.2019.07.025.
  6. Robert K. Brayton, Fred G. Gustavson, and Ralph A. Willoughby. Some results on sparse matrices. Math. Comp., 24:937-954, 1970. URL: https://doi.org/10.1090/s0025-5718-1970-0275643-8.
  7. Ke Chen. Matrix preconditioning techniques and applications, volume 19. Cambridge University Press, 2005. Google Scholar
  8. Richard Cole, Christoph Hertrich, Yixin Tao, and László A. Végh. A first order method for linear programming parameterized by circuit imbalance. CoRR, abs/2311.01959, 2023. URL: https://doi.org/10.48550/ARXIV.2311.01959.
  9. Daniel Dadush, Sophie Huiberts, Bento Natura, and László A. Végh. A scaling-invariant algorithm for linear programming whose running time depends only on the constraint matrix. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, pages 761-774, New York, NY, USA, June 2020. Association for Computing Machinery. Google Scholar
  10. Daniel Dadush, Zhuan Khye Koh, Bento Natura, and László A. Végh. On circuit diameter bounds via circuit imbalances. In Karen I. Aardal and Laura Sanità, editors, Integer Programming and Combinatorial Optimization - 23rd International Conference, IPCO 2022, Eindhoven, The Netherlands, June 27-29, 2022, Proceedings, volume 13265 of Lecture Notes in Computer Science, pages 140-153. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-06901-7_11.
  11. Farbod Ekbatani, Bento Natura, and László A Végh. Circuit imbalance measures and linear programming. Surveys in Combinatorics, pages 64-114, 2022. Google Scholar
  12. Bernd Gärtner and Jiří Matoušek. Understanding and using linear programming. Universitext. Springer, 2007. Google Scholar
  13. Gay, D. M. Electronic mail distribution of linear programming test problems. Mathematical Programming Society Committee on Algorithms Newsletter, 13, 1985. Data available from URL: https://netlib.org/lp/data/.
  14. Ambros M. Gleixner, Gregor Hendel, Gerald Gamrath, Tobias Achterberg, Michael Bastubbe, Timo Berthold, Philipp Christophel, Kati Jarck, Thorsten Koch, Jeff T. Linderoth, Marco E. Lübbecke, Hans D. Mittelmann, Derya B. Özyurt, Ted K. Ralphs, Domenico Salvagnin, and Yuji Shinano. MIPLIB 2017: data-driven compilation of the 6th mixed-integer programming library. Math. Program. Comput., 13(3):443-490, 2021. URL: https://doi.org/10.1007/s12532-020-00194-3.
  15. Gurobi Optimization, LLC. Gurobi Optimizer Reference Manual, 2023. URL: https://www.gurobi.com.
  16. Richard M. Karp. A characterization of the minimum cycle mean in a digraph. 23:309-311, 1978. URL: https://doi.org/10.1016/0012-365x(78)90011-0.
  17. Jakub Komárek. Experimental analysis of scaling methods for lp. 2023. Google Scholar
  18. Jesús A. De Loera, Sean Kafer, and Laura Sanità. Pivot rules for circuit-augmentation algorithms in linear optimization. SIAM J. Optim., 32(3):2156-2179, 2022. URL: https://doi.org/10.1137/21M1419994.
  19. István Maros. Computational techniques of the simplex method, volume 61. Springer Science & Business Media, 2002. Google Scholar
  20. J. G. Oxley. Matroid Theory. Oxford University Press, New York, 1992. Google Scholar
  21. The Sage Developers. SageMath, the Sage Mathematics Software System (Version 9.7), 2022. https://www.sagemath.org.
  22. Levent Tunçel. Approximating the complexity measure of vavasis-ye algorithm is np-hard. Math. Program., 86(1):219-223, 1999. URL: https://doi.org/10.1007/s101070050087.
  23. Stephen A. Vavasis and Yinyu Ye. A primal-dual interior point method whose running time depends only on the constraint matrix. 74:79-120, 1996. URL: https://doi.org/10.1007/bf02592148.