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Tight Lower Bounds for Block-Structured Integer Programs

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Integer Programming and Combinatorial Optimization (IPCO 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14679))

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Abstract

We study fundamental block-structured integer programs called tree-fold and multi-stage IPs. Tree-fold IPs admit a constraint matrix with independent blocks linked together by few constraints in a recursive pattern; and transposing their constraint matrix yields multi-stage IPs. The state-of-the-art algorithms to solve these IPs have an exponential gap in their running times, making it natural to ask whether this gap is inherent. We answer this question affirmative. Assuming the Exponential Time Hypothesis, we prove lower bounds showing that the exponential difference is necessary, and that the known algorithms are near optimal. Moreover, we prove unconditional lower bounds on the norms of the Graver basis, a fundamental building block of all known algorithms to solve these IPs. This shows that none of the current approaches can be improved beyond this bound.

C. Hunkenschröder acknowledges funding by Einstein Foundation Berlin. K.-M. Klein was supported by DFG project KL 3408/1-1. A. Lassota was partially supported by the Swiss National Science Foundation (SNSF) within the project Complexity of Integer Programming (207365). M. Koutecký was partially supported by the Charles University project UNCE 24/SCI/008 and by the project 22-22997S of GA ČR. A. Levin is partially supported by ISF – Israel Science Foundation grant number 1467/22.

A full version of this paper can be found in [14].

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References

  1. Albareda-Sambola, M., van der Vlerk, M.H., Fernández, E.: Exact solutions to a class of stochastic generalized assignment problems. Eur. J. Oper. Res. 173(2), 465–487 (2006)

    Article  MathSciNet  Google Scholar 

  2. Aschenbrenner, M., Hemmecke, R.: Finiteness theorems in stochastic integer programming. Found. Comput. Math. 7(2), 183–227 (2007)

    Article  MathSciNet  Google Scholar 

  3. Chen, L., Marx, D., Ye, D., Zhang, G.: Parameterized and approximation results for scheduling with a low rank processing time matrix. In: STACS. LIPIcs, vol. 66, pp. 22:1–22:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

    Google Scholar 

  4. Cslovjecsek, J., Eisenbrand, F., Hunkenschröder, C., Rohwedder, L., Weismantel, R.: Block-structured integer and linear programming in strongly polynomial and near linear time. In: SODA, pp. 1666–1681. SIAM (2021)

    Google Scholar 

  5. Cslovjecsek, J., Eisenbrand, F., Pilipczuk, M., Venzin, M., Weismantel, R.: Efficient sequential and parallel algorithms for multistage stochastic integer programming using proximity. In: ESA. LIPIcs, vol. 204, pp. 33:1–33:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

    Google Scholar 

  6. Cslovjecsek, J., Koutecký, M., Lassota, A., Pilipczuk, M., Polak, A.: Parameterized algorithms for block-structured integer programs with large entries. In: SODA 2024 (2024). https://arxiv.org/abs/2311.01890

  7. Dempster, M.A.H., Fisher, M.L., Jansen, L., Lageweg, B.J., Lenstra, J.K., Rinnooy Kan, A.H.G.: Analysis of heuristics for stochastic programming: results for hierarchical scheduling problems. Math. Oper. Res. 8(4), 525–537 (1983)

    Article  MathSciNet  Google Scholar 

  8. Eisenbrand, F., Hunkenschröder, C., Klein, K.-M.: Faster algorithms for integer programs with block structure. In: ICALP. LIPIcs, vol. 107, pp. 49:1–49:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

    Google Scholar 

  9. Eisenbrand, F., Hunkenschröder, C., Klein, K.-M., Kouteckỳ, M., Levin, A., Onn, S.: An algorithmic theory of integer programming. arXiv preprint arXiv:1904.01361 (2019)

  10. Eisenbrand, F., Weismantel, R.: Proximity results and faster algorithms for integer programming using the Steinitz lemma. ACM Trans. Algorithms (TALG) 16(1), 1–14 (2019)

    MathSciNet  Google Scholar 

  11. Gavenciak, T., Koutecký, M., Knop, D.: Integer programming in parameterized complexity: five miniatures. Discrete Optim. 44(Part), 100596 (2022)

    Article  MathSciNet  Google Scholar 

  12. Hemmecke, R., Onn, S., Romanchuk, L.: \(n\)-fold integer programming in cubic time. Math. Program. 137(1), 325–341 (2013)

    Article  MathSciNet  Google Scholar 

  13. Hemmecke, R., Onn, S., Weismantel, R.: \(n\)-fold integer programming and nonlinear multi-transshipment. Optim. Lett. 5(1), 13–25 (2011)

    Article  MathSciNet  Google Scholar 

  14. Hunkenschröder, C., Klein, K.-M., Koutecký, M., Lassota, A., Levin, A.: Tight lower bounds for block-structured integer programs (2024). https://arxiv.org/abs/2402.17290

  15. Jansen, K., Klein, K.-M., Lassota, A.: The double exponential runtime is tight for 2-stage stochastic ILPs. In: Singh, M., Williamson, D.P. (eds.) IPCO 2021. LNCS, vol. 12707, pp. 297–310. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-73879-2_21

    Chapter  Google Scholar 

  16. Jansen, K., Klein, K.-M., Maack, M., Rau, M.: Empowering the configuration-IP - new PTAS results for scheduling with setups times. In: ITCS. LIPIcs, vol. 124, pp. 44:1–44:19. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2019)

    Google Scholar 

  17. Jansen, K., Lassota, A., Maack, M., Pikies, T.: Total completion time minimization for scheduling with incompatibility cliques. In: ICAPS, pp. 192–200. AAAI Press (2021)

    Google Scholar 

  18. Jansen, K., Lassota, A., Rohwedder, L.: Near-linear time algorithm for \(n\)-fold ILPs via color coding. SIAM J. Discrete Math. 34(4), 2282–2299 (2020)

    Article  MathSciNet  Google Scholar 

  19. Kall, P., Wallace, S.W.: Stochastic Programming. Springer, Heidelberg (1994)

    Google Scholar 

  20. Klein, K.-M.: About the complexity of two-stage stochastic IPs. Math. Program. 192(1), 319–337 (2022)

    Article  MathSciNet  Google Scholar 

  21. Klein, K.-M., Reuter, J.: Collapsing the tower - on the complexity of multistage stochastic IPs (2022)

    Google Scholar 

  22. Knop, D., Koutecký, M.: Scheduling meets \(n\)-fold integer programming. J. Sched. 21(5), 493–503 (2018)

    Article  MathSciNet  Google Scholar 

  23. Knop, D., Koutecký, M.: Scheduling kernels via configuration LP. In: ESA. LIPIcs, vol. 244, pp. 73:1–73:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

    Google Scholar 

  24. Knop, D., Koutecký, M., Levin, A., Mnich, M., Onn, S.: Parameterized complexity of configuration integer programs. Oper. Res. Lett. 49(6), 908–913 (2021)

    Article  MathSciNet  Google Scholar 

  25. Knop, D., Koutecký, M., Mnich, M.: Combinatorial \(n\)-fold integer programming and applications. Math. Program. 184(1), 1–34 (2020)

    Article  MathSciNet  Google Scholar 

  26. Knop, D., Koutecký, M., Mnich, M.: Voting and bribing in single-exponential time. ACM Trans. Econ. Comput. 8(3), 12:1–12:28 (2020)

    Google Scholar 

  27. Knop, D., Pilipczuk, M., Wrochna, M.: Tight complexity lower bounds for integer linear programming with few constraints. ACM Trans. Comput. Theory 12(3), 1–19 (2020). https://doi.org/10.1145/3397484

    Article  MathSciNet  Google Scholar 

  28. Koutecký, M., Levin, A., Onn, S.: A parameterized strongly polynomial algorithm for block structured integer programs. In: 45th International Colloquium on Automata, Languages, and Programming. Leibniz International Proceedings in Informatics (LIPIcs), Germany, vol. 107, pp. 85:1–85:14. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik (2018)

    Google Scholar 

  29. Laporte, G., Louveaux, F.V., Mercure, H.: A priori optimization of the probabilistic traveling salesman problem. Oper. Res. 42(3), 543–549 (1994)

    Article  MathSciNet  Google Scholar 

  30. De Loera, J.A., Hemmecke, R., Onn, S., Weismantel, R.: N-fold integer programming. Discrete Optim. 5(2), 231–241 (2008)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Institut für Mathematik (formerly), TU Berlin, Berlin, Germany

    Christoph Hunkenschröder

  2. Institute for Theoretical Computer Science, University of Lübeck, Lübeck, Germany

    Kim-Manuel Klein

  3. Computer Science Institute, Charles University, Prague, Czech Republic

    Martin Koutecký

  4. Eindhoven University of Technology, Eindhoven, The Netherlands

    Alexandra Lassota

  5. Faculty of Data and Decisions Sciences, The Technion, Haifa, Israel

    Asaf Levin

Authors
  1. Christoph Hunkenschröder
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  2. Kim-Manuel Klein
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  3. Martin Koutecký
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  4. Alexandra Lassota
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  5. Asaf Levin
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Corresponding author

Correspondence to Alexandra Lassota .

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Editors and Affiliations

  1. University of Bonn, Bonn, Germany

    Jens Vygen

  2. University of Wrocław, Wrocław, Poland

    Jarosław Byrka

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Hunkenschröder, C., Klein, KM., Koutecký, M., Lassota, A., Levin, A. (2024). Tight Lower Bounds for Block-Structured Integer Programs. In: Vygen, J., Byrka, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2024. Lecture Notes in Computer Science, vol 14679. Springer, Cham. https://doi.org/10.1007/978-3-031-59835-7_17

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  • DOI: https://doi.org/10.1007/978-3-031-59835-7_17

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