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A Brief History of Parameterized Algorithms for Block-Structured Integer Programs (Invited Talk)

Author Martin Koutecký

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  • This paper is partially based on my habilitation. It contains some material which was omitted here due to space limitations, in particular an entire chapter on the papers [Briański et al., 2024; Timothy F. N. Chan et al., 2022].
  • Habilitation https://research.koutecky.name/papers/habil.pdf

Subject Classification

ACM Subject Classification
  • Theory of computation → Convex optimization
Keywords
  • Integer Programming
  • Parameterized Algorithm
  • Graver Basis
  • Treedepth
  • n-fold
  • tree-fold
  • 2-stage stochastic
  • multistage stochastic
  • Mixed-Integer Programming

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Abstract

Integer Programming (IP) is a fundamental but computationally hard problem. Still, certain efficiently solvable subclasses have been identified over time, most notably totally unimodular IPs in the 1950s, and fixed-dimension IPs in the 1980s. Starting around the year 2000, a stream of research has identified block-structured IPs as yet another tractable subclass. In this paper, we give a brief and incomplete review of this history, with a focus on several of the author’s contributions.

Cite As Get BibTex

Martin Koutecký. A Brief History of Parameterized Algorithms for Block-Structured Integer Programs (Invited Talk). In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 1:1-1:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.IPEC.2025.1

Author Details

Martin Koutecký
  • Computer Science Institute, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

Funding

Partially supported by Charles Univ. project UNCE 24/SCI/008, by the ERC-CZ project LL2406 of the Ministry of Education of Czech Republic, and by the project 24-10306S of GA ČR.

References

  1. Sibel A. Alumur and Bahar Yetis Kara. Network hub location problems: The state of the art. European Journal of Operational Research, 190(1):1-21, 2008. URL: https://doi.org/10.1016/J.EJOR.2007.06.008.
  2. Gautam Appa, Balázs Kotnyek, Konstantinos Papalamprou, and Leonidas Pitsoulis. Optimization with binet matrices. Operations research letters, 35(3):345-352, 2007. URL: https://doi.org/10.1016/J.ORL.2006.04.003.
  3. Stephan Artmann, Robert Weismantel, and Rico Zenklusen. A strongly polynomial algorithm for bimodular integer linear programming. In STOC 2017, pages 1206-1219. ACM, 2017. URL: https://doi.org/10.1145/3055399.3055473.
  4. Matthias Aschenbrenner and Raymond Hemmecke. Finiteness theorems in stochastic integer programming. Foundations of Computational Mathematics, 7(2):183-227, 2007. URL: https://doi.org/10.1007/S10208-005-0174-1.
  5. Jacques F. Benders. Partitioning procedures for solving mixed-variables programming problems. Comput. Manag. Sci., 2(1):3-19, 2005. URL: https://doi.org/10.1007/S10287-004-0020-Y.
  6. Dimitris Bertsimas, Angela King, and Rahul Mazumder. Best subset selection via a modern optimization lens. The Annals of Statistics, 44(2):813-852, 2016. URL: http://www.jstor.org/stable/43818629.
  7. John R. Birge and François Louveaux. Introduction to stochastic programming. Springer-Verlag, New York, 1997. Google Scholar
  8. Robert E Bixby. Solving real-world linear programs: A decade and more of progress. Operations research, 50(1):3-15, 2002. URL: https://doi.org/10.1287/OPRE.50.1.3.17780.
  9. Cornelius Brand, Martin Koutecký, Alexandra Lassota, and Sebastian Ordyniak. Separable Convex Mixed-Integer Optimization: Improved Algorithms and Lower Bounds. In ESA 2024, volume 308 of LIPIcs, pages 32:1-32:18, Dagstuhl, Germany, 2024. URL: https://doi.org/10.4230/LIPIcs.ESA.2024.32.
  10. Cornelius Brand, Martin Koutecký, and Sebastian Ordyniak. Parameterized algorithms for MILPs with small treedepth. In AAAI 2021, pages 12249-12257. AAAI Press, 2021. URL: https://doi.org/10.1609/AAAI.V35I14.17454.
  11. Marcin Briański, Martin Koutecký, Daniel Kráľ, Kristýna Pekárková, and Felix Schröder. Characterization of matrices with bounded graver bases and depth parameters and applications to integer programming. Mathematical Programming, pages 1-35, 2024. URL: https://doi.org/10.1007/s10107-023-02048-x.
  12. Hauke Brinkop and Klaus Jansen. High multiplicity scheduling on uniform machines in fpt-time. CoRR, abs/2203.01741, 2022. URL: https://doi.org/10.48550/arXiv.2203.01741.
  13. Timothy F. N. Chan, Jacob W. Cooper, Martin Koutecký, Daniel Kráľ, and Kristýna Pekárková. Matrices of optimal tree-depth and a row-invariant parameterized algorithm for integer programming. SIAM J. Comput., 51(3):664-700, 2022. URL: https://doi.org/10.1137/20m1353502.
  14. Lin Chen and Dániel Marx. Covering a tree with rooted subtrees-parameterized and approximation algorithms. In SODA 2018, pages 2801-2820. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.178.
  15. Lin Chen, Dániel Marx, Deshi Ye, and Guochuan Zhang. Parameterized and approximation results for scheduling with a low rank processing time matrix. In STACS 2017, volume 66 of LIPIcs, pages 22:1-22:14, 2017. URL: https://doi.org/10.4230/LIPICS.STACS.2017.22.
  16. Sergei Chubanov. A polynomial-time descent method for separable convex optimization problems with linear constraints. SIAM Journal on Optimization, 26(1):856-889, 2016. URL: https://doi.org/10.1137/14098524X.
  17. Michele Conforti, Marco Di Summa, Friedrich Eisenbrand, and Laurence A. Wolsey. Network formulations of mixed-integer programs. Math. Oper. Res., 34(1):194-209, 2009. URL: https://doi.org/10.1287/moor.1080.0354.
  18. Stavros S. Cosmadakis and Christos H. Papadimitriou. The traveling salesman problem with many visits to few cities. SIAM J. Comput., 13(1):99-108, 1984. URL: https://doi.org/10.1137/0213007.
  19. Jana Cslovjecsek, Friedrich Eisenbrand, Christoph Hunkenschröder, Lars Rohwedder, and Robert Weismantel. Block-structured integer and linear programming in strongly polynomial and near linear time. In SODA 2021, pages 1666-1681. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.101.
  20. Jana Cslovjecsek, Friedrich Eisenbrand, Michał Pilipczuk, Moritz Venzin, and Robert Weismantel. Efficient sequential and parallel algorithms for multistage stochastic integer programming using proximity. In ESA 2021, volume 204 of LIPIcs, pages 33:1-33:14, 2021. URL: https://doi.org/10.4230/LIPIcs.ESA.2021.33.
  21. Jana Cslovjecsek, Martin Koutecký, Alexandra Lassota, Michał Pilipczuk, and Adam Polak. Parameterized algorithms for block-structured integer programs with large entries. In SODA 2024, pages 740-751. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.29.
  22. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  23. George B Dantzig. Application of the simplex method to a transportation problem. Activity Analysis and Production and Allocation, 1951. Google Scholar
  24. George B Dantzig and Philip Wolfe. Decomposition principle for linear programs. Operations research, 8(1):101-111, 1960. Google Scholar
  25. Jesús A. De Loera, Raymond Hemmecke, and Jon Lee. On augmentation algorithms for linear and integer-linear programming: From Edmonds-Karp to Bland and beyond. SIAM Journal on Optimization, 25(4):2494-2511, 2015. URL: https://doi.org/10.1137/151002915.
  26. Jesús A. De Loera, Raymond Hemmecke, Shmuel Onn, and Robert Weismantel. N-fold integer programming. Discrete Optimization, 5(2):231-241, 2008. URL: https://doi.org/10.1016/J.DISOPT.2006.06.006.
  27. Jesús A. De Loera and Shmuel Onn. The complexity of three-way statistical tables. SIAM J. Comput, 33(4):819-836, 2004. URL: https://doi.org/10.1137/S0097539702403803.
  28. Jesús A. De Loera and Shmuel Onn. All linear and integer programs are slim 3-way transportation programs. SIAM Journal on Optimization, 17(3):806-821, 2006. URL: https://doi.org/10.1137/040610623.
  29. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL: https://doi.org/10.1007/978-1-4471-5559-1.
  30. Pavel Dvořák, Eduard Eiben, Robert Ganian, Dušan Knop, and Sebastian Ordyniak. The complexity landscape of decompositional parameters for ILP: programs with few global variables and constraints. Artif. Intell., 300:103561, 2021. URL: https://doi.org/10.1016/j.artint.2021.103561.
  31. Friedrich Eisenbrand, Christoph Hunkenschröder, Kim-Manuel Klein, Martin Koutecký, Asaf Levin, and Shmuel Onn. Sparse integer programming is fixed-parameter tractable. Mathematics of Operations Research, 0(0):null, 0. URL: https://doi.org/10.1287/moor.2023.0162.
  32. Friedrich Eisenbrand, Christoph Hunkenschröder, Kim-Manuel Klein, Martin Koutecký, Asaf Levin, and Shmuel Onn. Reducibility bounds of objective functions over the integers. Oper. Res. Lett., 51(6):595-598, 2023. URL: https://doi.org/10.1016/J.ORL.2023.10.001.
  33. Friedrich Eisenbrand, Christoph Hunkenschröder, Kim-Manuel Klein, Martin Koutecký, Asaf Levin, and Shmuel Onn. An algorithmic theory of integer programming, 2019. URL: https://arxiv.org/abs/1904.01361.
  34. Friedrich Eisenbrand and Thomas Rothvoss. A parameterized linear formulation of the integer hull. CoRR, abs/2501.02347, 2025. URL: https://doi.org/10.48550/arXiv.2501.02347.
  35. Farbod Ekbatani, Bento Natura, and László A. Végh. Circuit imbalance measures and linear programming. CoRR, abs/2108.03616, 2021. URL: https://arxiv.org/abs/2108.03616.
  36. Piotr Faliszewski, Rica Gonen, Martin Koutecký, and Nimrod Talmon. Opinion diffusion and campaigning on society graphs. J. Log. Comput., 32(6):1162-1194, 2022. URL: https://doi.org/10.1093/logcom/exac014.
  37. Stephen E Fienberg and Alessandro Rinaldo. Three centuries of categorical data analysis: Log-linear models and maximum likelihood estimation. Journal of Statistical Planning and Inference, 137(11):3430-3445, 2007. Google Scholar
  38. Samuel Fiorini, Gwenaël Joret, Stefan Weltge, and Yelena Yuditsky. Integer programs with bounded subdeterminants and two nonzeros per row. In FOCS 2021, pages 13-24. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00011.
  39. Christodoulos A. Floudas and Xiaoxia Lin. Mixed integer linear programming in process scheduling: Modeling, algorithms, and applications. Ann. Oper. Res., 139(1):131-162, 2005. URL: https://doi.org/10.1007/S10479-005-3446-X.
  40. András Frank and Éva Tardos. An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica, 7(1):49-65, 1987. URL: https://doi.org/10.1007/BF02579200.
  41. Eugene C. Freuder. Complexity of K-tree structured constraint satisfaction problems. In Proceedings of the 8th National Conference on Artificial Intelligence, pages 4-9, 1990. URL: http://www.aaai.org/Library/AAAI/1990/aaai90-001.php.
  42. Robert Ganian and Sebastian Ordyniak. The complexity landscape of decompositional parameters for ILP. Artificial Intelligence, 2018. Google Scholar
  43. Robert Ganian, Sebastian Ordyniak, and M. S. Ramanujan. Going beyond primal treewidth for (M)ILP. In AAAI, pages 815-821, 2017. URL: https://doi.org/10.1609/AAAI.V31I1.10644.
  44. Tomáš Gavenčiak, Martin Koutecký, and Dušan Knop. Integer programming in parameterized complexity: Five miniatures. Discret. Optim., 44(Part):100596, 2022. URL: https://doi.org/10.1016/j.disopt.2020.100596.
  45. Paul C Gilmore and Ralph E Gomory. A linear programming approach to the cutting-stock problem. Operations research, 9(6):849-859, 1961. Google Scholar
  46. Paul C Gilmore and Ralph E Gomory. A linear programming approach to the cutting stock problem - part ii. Operations research, 11(6):863-888, 1963. Google Scholar
  47. Fred W. Glover. Tabu search - part II. INFORMS J. Comput., 2(1):4-32, 1990. URL: https://doi.org/10.1287/IJOC.2.1.4.
  48. Dmitry V. Gribanov, Dmitry S. Malyshev, and Ivan A. Shumilov. On a simple connection between Δ-modular ILP and lp, and a new bound on the number of integer vertices. Oper. Res. Forum, 5(2):32, 2024. URL: https://doi.org/10.1007/S43069-024-00310-2.
  49. Martin Grötschel, László Lovász, and Alexander Schrijver. Geometric algorithms and combinatorial optimization, volume 2 of Algorithms and Combinatorics. Springer-Verlag, Berlin, second edition, 1993. Google Scholar
  50. Claire Hanen and Alix Munier Kordon. Fixed-parameter tractability of scheduling dependent typed tasks subject to release times and deadlines. J. Sched., 27(2):119-133, 2024. URL: https://doi.org/10.1007/S10951-023-00788-4.
  51. Raymond Hemmecke, Matthias Köppe, and Robert Weismantel. Graver basis and proximity techniques for block-structured separable convex integer minimization problems. Mathematical Programming, 145(1-2, Ser. A):1-18, 2014. URL: https://doi.org/10.1007/S10107-013-0638-Z.
  52. Raymond Hemmecke, Shmuel Onn, and Lyubov Romanchuk. N-fold integer programming in cubic time. Mathematical Programming, pages 1-17, 2013. Google Scholar
  53. Raymond Hemmecke and Rüdiger Schultz. Decomposition of test sets in stochastic integer programming. Mathematical Programming, 94:323-341, 2003. URL: https://doi.org/10.1007/S10107-002-0322-1.
  54. Frank L Hitchcock. The distribution of a product from several sources to numerous localities. Journal of mathematics and physics, 20(1-4):224-230, 1941. Google Scholar
  55. Dorit S. Hochbaum and Ron Shamir. Strongly polynomial algorithms for the high multiplicity scheduling problem. Oper. Res., 39(4):648-653, 1991. URL: https://doi.org/10.1287/OPRE.39.4.648.
  56. Dorit S. Hochbaum and J. George Shanthikumar. Convex separable optimization is not much harder than linear optimization. Journal of the ACM, 37(4):843-862, 1990. URL: https://doi.org/10.1145/96559.96597.
  57. Alan J Hoffman and Joseph B Kruskal. Integral boundary points of convex polyhedra. Linear inequalities and related systems, pages 223-246, 1956. Google Scholar
  58. Serkan Hoşten and Seth Sullivant. A finiteness theorem for markov bases of hierarchical models. J. Comb. Theory A, 114(2):311-321, 2007. URL: https://doi.org/10.1016/J.JCTA.2006.06.001.
  59. Bart M. P. Jansen and Stefan Kratsch. A structural approach to kernels for ILPs: Treewidth and total unimodularity. In ESA, volume 9294 of Lecture Notes in Computer Science, pages 779-791, 2015. URL: https://doi.org/10.1007/978-3-662-48350-3_65.
  60. Klaus Jansen, Kim-Manuel Klein, Marten Maack, and Malin Rau. Empowering the configuration-IP: new PTAS results for scheduling with setup times. Math. Program., 195(1):367-401, 2022. URL: https://doi.org/10.1007/S10107-021-01694-3.
  61. Peter Kall and Stein W. Wallace. Stochastic Programming. Wiley, Chichester etc., 1994. Google Scholar
  62. Ravi Kannan. Minkowski’s convex body theorem and integer programming. Mathematics of Operations Research, 12(3):415-440, August 1987. URL: https://doi.org/10.1287/MOOR.12.3.415.
  63. Leonid V Kantorovich. On the translocation of masses. In Dokl. Akad. Nauk. USSR (NS), volume 37, pages 199-201, 1942. Google Scholar
  64. Richard M. Karp. Reducibility among combinatorial problems. In Complexity of Computer Computations, The IBM Research Symposia Series, pages 85-103. Springer, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  65. Kim-Manuel Klein, Adam Polak, and Lars Rohwedder. On minimizing tardy processing time, max-min skewed convolution, and triangular structured ILPs. In SODA 2023, pages 2947-2960, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH112.
  66. Dušan Knop, Martin Koutecký, Asaf Levin, Matthias Mnich, and Shmuel Onn. High-multiplicity N-fold IP via configuration LP. Mathematical Programming, 200(1):199-227, June 2023. URL: https://doi.org/10.1007/s10107-022-01882-9.
  67. Dušan Knop and Martin Koutecký. Scheduling meets n-fold integer programming. J. Scheduling, 21(5):493-503, 2018. URL: https://doi.org/10.1007/S10951-017-0550-0.
  68. Dušan Knop and Martin Koutecký. Scheduling kernels via configuration LP. In ESA 2022, volume 244 of LIPIcs, pages 73:1-73:15, 2022. URL: https://doi.org/10.4230/LIPICS.ESA.2022.73.
  69. Dušan Knop, Martin Koutecký, Asaf Levin, Matthias Mnich, and Shmuel Onn. Multitype integer monoid optimization and applications. CoRR, abs/1909.07326, 2019. URL: https://arxiv.org/abs/1909.07326.
  70. Dušan Knop, Martin Koutecký, and Matthias Mnich. A unifying framework for manipulation problems. In AAMAS 2018, pages 256-264. International Foundation for Autonomous Agents and Multiagent Systems Richland, SC, USA / ACM, 2018. URL: http://dl.acm.org/citation.cfm?id=3237427.
  71. Dušan Knop, Martin Koutecký, and Matthias Mnich. Combinatorial n-fold integer programming and applications. Math. Program., 184(1):1-34, 2020. URL: https://doi.org/10.1007/s10107-019-01402-2.
  72. Dušan Knop, Martin Koutecký, and Matthias Mnich. Voting and bribing in single-exponential time. ACM Trans. Economics and Comput., 8(3):12:1-12:28, 2020. URL: https://doi.org/10.1145/3396855.
  73. Martin Koutecký, Asaf Levin, and Shmuel Onn. A parameterized strongly polynomial algorithm for block structured integer programs. In Proc. ICALP 2018, volume 107 of LIPIcs, pages 85:1-85:14, 2018. URL: https://doi.org/10.4230/LIPICS.ICALP.2018.85.
  74. Martin Koutecký and Nimrod Talmon. Multi-party campaigning. In AAAI 2021, pages 5506-5513. AAAI Press, 2021. URL: https://doi.org/10.1609/AAAI.V35I6.16693.
  75. Hendrik W. Lenstra, Jr. Integer programming with a fixed number of variables. Mathematics of Operations Research, 8(4):538-548, 1983. URL: https://doi.org/10.1287/MOOR.8.4.538.
  76. Andrea Lodi, Silvano Martello, and Michele Monaci. Two-dimensional packing problems: A survey. European Journal of Operational Research, 141(2):241-252, 2002. URL: https://doi.org/10.1016/S0377-2217(02)00123-6.
  77. Diane Maclagan. Antichains of monomial ideals are finite. Proceedings of the American Mathematical Society, 129(6):1609-1615, 2001. Google Scholar
  78. Hugues Marchand, Alexander Martin, Robert Weismantel, and Laurence A. Wolsey. Cutting planes in integer and mixed integer programming. Discret. Appl. Math., 123(1-3):397-446, 2002. URL: https://doi.org/10.1016/S0166-218X(01)00348-1.
  79. Alexander Martin. Integer programs with block structure. PhD thesis, Zuse Institute Berlin, 1999. Google Scholar
  80. Theodore Samuel Motzkin. The multi-index transportation problem. In Bulletin of the American Mathematical Society, volume 58(4), pages 494-494, 1952. Google Scholar
  81. Martin Nägele, Richard Santiago, and Rico Zenklusen. Congruency-constrained TU problems beyond the bimodular case. In SODA 2022, pages 2743-2790. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.108.
  82. Jaroslav Nešetřil and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-27875-4.
  83. Ivo Nowak. Relaxation and Decomposition Methods for Mixed Integer Nonlinear Programming, Second Edition, volume 152. Birkhäuser Basel, 2005. URL: https://doi.org/10.1007/3-7643-7374-1.
  84. John E Olson. A combinatorial problem on finite abelian groups, I. Journal of number theory, 1(1):8-10, 1969. Google Scholar
  85. Joseph Paat, Robert Weismantel, and Stefan Weltge. Distances between optimal solutions of mixed-integer programs. Math. Program., 179(1):455-468, 2020. URL: https://doi.org/10.1007/s10107-018-1323-z.
  86. Christos H. Papadimitriou. On the complexity of integer programming. J. ACM, 28(4):765-768, 1981. URL: https://doi.org/10.1145/322276.322287.
  87. Marcin Pilipczuk, Michał Pilipczuk, and Sebastian Siebertz. Lecture notes from the course "Sparsity" given at the Faculty of Mathematics, Informatics, and Mechanics of the University of Warsaw, Winter semesters 2017/18 and 2019/20. Available online at URL: https://www.mimuw.edu.pl/~mp248287/sparsity2.
  88. András Prékopa. Stochastic programming, volume 324 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1995. Google Scholar
  89. Harilaos N. Psaraftis. A dynamic programming approach for sequencing groups of identical jobs. Oper. Res., 28(6):1347-1359, 1980. URL: https://doi.org/10.1287/OPRE.28.6.1347.
  90. Victor Reis and Thomas Rothvoss. The subspace flatness conjecture and faster integer programming. In FOCS 2023, pages 974-988. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00060.
  91. Francisco Santos and Bernd Sturmfels. Higher lawrence configurations. J. Comb. Theory A, 103(1):151-164, 2003. URL: https://doi.org/10.1016/S0097-3165(03)00092-X.
  92. Guntram Scheithauer and Johannes Terno. The modified integer round-up property of the one-dimensional cutting stock problem. European J. Oper. Res., 84(3):562-571, 1995. Google Scholar
  93. Éva Tardos. A strongly polynomial algorithm to solve combinatorial linear programs. Operations Research, 34(2):250-256, 1986. URL: https://doi.org/10.1287/OPRE.34.2.250.
  94. D. Antony Tarvin, R. Kevin Wood, and Alexandra M. Newman. Benders decomposition: Solving binary master problems by enumeration. Oper. Res. Lett., 44(1):80-85, 2016. URL: https://doi.org/10.1016/J.ORL.2015.11.009.
  95. Paolo Toth and Daniele Vigo, editors. The Vehicle Routing Problem. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2001. Google Scholar
  96. J. M. van den Akker, J. A. Hoogeveen, and Steef L. van de Velde. Parallel machine scheduling by column generation. Oper. Res., 47(6):862-872, 1999. URL: https://doi.org/10.1287/OPRE.47.6.862.
  97. Menkes van den Briel, Thomas Vossen, and Subbarao Kambhampati. Reviving integer programming approaches for AI planning: A branch-and-cut framework. In Automated Planning and Scheduling, ICAPS 2005. Proceedings, pages 310-319. AAAI, 2005. URL: http://www.aaai.org/Library/ICAPS/2005/icaps05-032.php.
  98. François Vanderbeck and Martin W. P. Savelsbergh. A generic view of Dantzig-Wolfe decomposition in mixed integer programming. Oper. Res. Lett., 34(3):296-306, 2006. URL: https://doi.org/10.1016/J.ORL.2005.05.009.
  99. Laurence A Wolsey. Integer programming. John Wiley & Sons, 2020. Google Scholar
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