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Adaptive Sparsification for Linear Programming

Authors Etienne Objois , Adrian Vladu

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LIPIcs.ESA.2026.40.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • linear programming
  • sparsification
  • sampling
  • quantum algorithms

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Abstract

We provide a generic toolkit for sparsifying the constraint set of linear programs (LPs). To this end, we reduce solving a linear program with n constraints and d variables (n≫ d), to solving a sequence of LPs defined over only a small subset of the constraints, obtained by adaptively sub-sampling the original set. We provide results for both the low and high precision regimes. To achieve the former result, we streamline and generalize the techniques from [Assadi '25] for approximately computing maximum matchings in the semi-streaming setting to the case of general LPs. For the latter, we robustify the methods of [Clarkson '95], which were originally designed for exact LP solvers. As a consequence we obtain fast approximate LP solvers which reduce the dependence on width and error from quadratic to linear, compared to vanilla multiplicative-weights based approaches.
Additionally, we leverage our findings to obtain fast LP solvers in the quantum query access model, where the running time scales with √n. This completely decouples the component responsible for quantum speed-ups, solely represented by a generalization of Grover’s search, from its classical algorithmic counterpart.

Cite As Get BibTex

Etienne Objois and Adrian Vladu. Adaptive Sparsification for Linear Programming. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 40:1-40:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.40

Author Details

Etienne Objois
  • IRIF, Paris, France
  • Université Paris Cité, France
Adrian Vladu
  • CNRS, Paris, France
  • IRIF, Paris, France
  • Université Paris Cité, France

Funding

This work was partially supported by the French Agence Nationale de la Recherche (ANR) under grant ANR-21-CE48-0016 (project COMCOPT).

References

  1. Zeyuan Allen-Zhu and Lorenzo Orecchia. Nearly linear-time packing and covering LP solvers. Math. Program., 175(1-2):307-353, May 2019. URL: https://doi.org/10.1007/s10107-018-1244-x.
  2. Simon Apers and Ronald de Wolf. Quantum Speedup for Graph Sparsification, Cut Approximation, and Laplacian Solving. SIAM Journal on Computing, 2023. Google Scholar
  3. Simon Apers and Sander Gribling. Quantum Speedups for Linear Programming via Interior Point Methods. SIAM Journal on Computing, 55(1):93-134, February 2026. URL: https://doi.org/10.1137/25M1736098.
  4. Sanjeev Arora, Elad Hazan, and Satyen Kale. The Multiplicative Weights Update Method: A Meta-Algorithm and Applications. Theory of Computing, 8(6):121-164, May 2012. URL: https://doi.org/10.4086/toc.2012.v008a006.
  5. Sepehr Assadi. A Simple (1-ε)-Approximation Semi-Streaming Algorithm for Maximum (Weighted) Matching. TheoretiCS, Volume 4, August 2025. URL: https://doi.org/10.46298/theoretics.25.16.
  6. András A. Benczúr and David R. Karger. Randomized Approximation Schemes for Cuts and Flows in Capacitated Graphs. SIAM Journal on Computing, 44(2):290-319, January 2015. URL: https://doi.org/10.1137/070705970.
  7. Digvijay Boob, Saurabh Sawlani, and Di Wang. Faster width-dependent algorithm for mixed packing and covering LPs. In Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019. Google Scholar
  8. Chandra Chekuri and Kent Quanrud. Randomized MWU for Positive LPs. In Proceedings of the 2018 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Proceedings, pages 358-377. Society for Industrial and Applied Mathematics, January 2018. URL: https://doi.org/10.1137/1.9781611975031.25.
  9. Kenneth L. Clarkson. Las Vegas algorithms for linear and integer programming when the dimension is small. J. ACM, 42(2):488-499, March 1995. URL: https://doi.org/10.1145/201019.201036.
  10. Michael B. Cohen, Yin Tat Lee, and Zhao Song. Solving linear programs in the current matrix multiplication time. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, pages 938-942, New York, NY, USA, June 2019. Association for Computing Machinery. URL: https://doi.org/10.1145/3313276.3316303.
  11. Michael D. Grigoriadis and Leonid G. Khachiyan. A sublinear-time randomized approximation algorithm for matrix games. Operations Research Letters, 18(2):53-58, September 1995. URL: https://doi.org/10.1016/0167-6377(95)00032-0.
  12. Yin Tat Lee and Aaron Sidford. Efficient Inverse Maintenance and Faster Algorithms for Linear Programming. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 230-249, October 2015. URL: https://doi.org/10.1109/FOCS.2015.23.
  13. Yin Tat Lee, Aaron Sidford, and Sam Chiu-Wai Wong. A Faster Cutting Plane Method and its Implications for Combinatorial and Convex Optimization. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 1049-1065, October 2015. URL: https://doi.org/10.1109/FOCS.2015.68.
  14. S.A. Plotkin, D.B. Shmoys, and E. Tardos. Fast approximation algorithms for fractional packing and covering problems. In [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science, pages 495-504, October 1991. URL: https://doi.org/10.1109/SFCS.1991.185411.
  15. Kent Quanrud. Nearly linear time approximations for mixed packing and covering problems without data structures or randomization. In 2020 Symposium on Simplicity in Algorithms (SOSA), Proceedings, pages 69-80. Society for Industrial and Applied Mathematics, December 2019. URL: https://doi.org/10.1137/1.9781611976014.11.
  16. Kent Quanrud. Adaptive Sparsification for Matroid Intersection. LIPIcs, Volume 297, ICALP 2024, 297:118:1-118:20, 2024. URL: https://doi.org/10.4230/LIPIcs.ICALP.2024.118.
  17. Jonah Sherman. Area-convexity, 𝓁_∞ regularization, and undirected multicommodity flow. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pages 452-460, New York, NY, USA, June 2017. Association for Computing Machinery. URL: https://doi.org/10.1145/3055399.3055501.
  18. Steve Smale. Mathematical problems for the next century. The Mathematical Intelligencer, 20(2):7-15, March 1998. URL: https://doi.org/10.1007/BF03025291.
  19. Daniel A. Spielman and Nikhil Srivastava. Graph sparsification by effective resistances. In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, STOC '08, pages 563-568, New York, NY, USA, May 2008. Association for Computing Machinery. URL: https://doi.org/10.1145/1374376.1374456.
  20. P.M. Vaidya. Speeding-up linear programming using fast matrix multiplication. In 30th Annual Symposium on Foundations of Computer Science, pages 332-337, October 1989. URL: https://doi.org/10.1109/SFCS.1989.63499.
  21. Joran van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf. Quantum SDP-Solvers: Better upper and lower bounds. Quantum, 4:230, February 2020. URL: https://doi.org/10.22331/q-2020-02-14-230.
  22. Jan van den Brand, Yin Tat Lee, Aaron Sidford, and Zhao Song. Solving tall dense linear programs in nearly linear time. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, pages 775-788, New York, NY, USA, June 2020. Association for Computing Machinery. URL: https://doi.org/10.1145/3357713.3384309.
  23. Di Wang, Satish Rao, and Michael W. Mahoney. Unified Acceleration Method for Packing and Covering Problems via Diameter Reduction. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016), volume 55 of Leibniz International Proceedings in Informatics (LIPIcs), pages 50:1-50:13, Dagstuhl, Germany, 2016. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2016.50.
  24. Neal E. Young. Nearly Linear-Work Algorithms for Mixed Packing/Covering and Facility-Location Linear Programs, November 2014. URL: https://arxiv.org/abs/1407.3015.
  25. Étienne Objois and Adrian Vladu. Adaptive sparsification for linear programming, 2025. URL: https://doi.org/10.48550/arXiv.2510.08348.
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