New Support Size Bounds for Integer Programming, Applied to Makespan Minimization on Uniformly Related Machines

Authors Sebastian Berndt , Hauke Brinkop , Klaus Jansen , Matthias Mnich , Tobias Stamm

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Author Details

Sebastian Berndt
  • Institute for Theoretical Computer Science, University of Lübeck, Germany
Hauke Brinkop
  • Kiel University, Germany
Klaus Jansen
  • Kiel University, Germany
Matthias Mnich
  • Hamburg University of Technology, Institute for Algorithms and Complexity, Germany
Tobias Stamm
  • Hamburg University of Technology, Institute for Algorithms and Complexity, Germany

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Sebastian Berndt, Hauke Brinkop, Klaus Jansen, Matthias Mnich, and Tobias Stamm. New Support Size Bounds for Integer Programming, Applied to Makespan Minimization on Uniformly Related Machines. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Mixed-integer linear programming (MILP) is at the core of many advanced algorithms for solving fundamental problems in combinatorial optimization. The complexity of solving MILPs directly correlates with their support size, which is the minimum number of non-zero integer variables in an optimal solution. A hallmark result by Eisenbrand and Shmonin (Oper. Res. Lett. , 2006) shows that any feasible integer linear program (ILP) has a solution with support size s ≤ 2m⋅log(4mΔ), where m is the number of constraints, and Δ is the largest absolute coefficient in any constraint. Our main combinatorial result are improved support size bounds for ILPs. We show that any ILP has a solution with support size s ≤ m⋅(log(3A_max)+√{log(A_max)}), where A_max≔ ‖A‖₁ denotes the 1-norm of the constraint matrix A. Furthermore, we show support bounds in the linearized form s ≤ 2m⋅log(1.46 A_max). Our upper bounds also hold with A_max replaced by √mΔ, which improves on the previously best constants in the linearized form. Our main algorithmic result are the fastest known approximation schemes for fundamental scheduling problems, which use the improved support bounds as one ingredient. We design an efficient approximation scheme (EPTAS) for makespan minimization on uniformly related machines (Q||C_{max}). Our EPTAS yields a (1+ε)-approximation for Q||C_{max} on N jobs in time 2^𝒪(1/ε log³(1/ε)log(log(1/ε))) + 𝒪(N), which improves over the previously fastest algorithm by Jansen, Klein and Verschae (Math. Oper. Res., 2020) with run time 2^𝒪(1/ε log⁴(1/ε)) + N^𝒪(1). Arguably, our approximation scheme is also simpler than all previous EPTASes for Q||C_max, as we reduce the problem to a novel MILP formulation which greatly benefits from the small support.

Subject Classification

ACM Subject Classification
  • Theory of computation → Scheduling algorithms
  • Integer programming
  • scheduling algorithms
  • uniformly related machines
  • makespan minimization


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  1. Iskander Aliev, Gennadiy Averkov, Jesús A. De Loera, and Timm Oertel. Sparse representation of vectors in lattices and semigroups. Math. Program., 192(1-2, Ser. B):519-546, 2022. URL:
  2. Iskander Aliev, Jesús A. De Loera, Friedrich Eisenbrand, T. Oertel, and Robert Weismantel. The support of integer optimal solutions. SIAM J. Optim., 28(3):2152-2157, 2018. URL:
  3. Iskander Aliev, Jesús A. De Loera, Timm Oertel, and Christopher O'Neill. Sparse solutions of linear Diophantine equations. SIAM J. Appl. Algebra Geom., 1(1):239-253, 2017. URL:
  4. Yossi Azar and Leah Epstein. Approximation schemes for covering and scheduling in related machines. Proc. APPROX 1998, 1444:39-47, 1998. URL:
  5. Nikhil Bansal, Tim Oosterwijk, Tjark Vredeveld, and Ruben van der Zwaan. Approximating vector scheduling: almost matching upper and lower bounds. Algorithmica, 76(4):1077-1096, 2016. URL:
  6. Heinz Bauer. Minimalstellen von Funktionen und Extremalpunkte. II. Arch. Math., 11:200-205, 1960. URL:
  7. Sebastian Berndt, Hauke Brinkop, Klaus Jansen, Matthias Mnich, and Tobias Stamm. New support size bounds for integer programming, applied to makespan minimization on uniformly related machines, 2023. URL:
  8. Sebastian Berndt, Max A. Deppert, Klaus Jansen, and Lars Rohwedder. Load balancing: The long road from theory to practice. In Proc. ALENEX 2022, pages 104-116, 2022. URL:
  9. Sebastian Berndt, Klaus Jansen, and Kim-Manuel Klein. New bounds for the vertices of the integer hull. Proc. SODA 2021, pages 25-36, 2021. URL:
  10. Patrick Browne, Ronan Egan, Fintan Hegarty, and Padraig Ó Catháin. A survey of the Hadamard maximal determinant problem. Electron. J. Combin., 28(4):Paper No. 4.41,35, 2021. URL:
  11. Ioannis Chatzigeorgiou. Bounds on the Lambert function and their application to the outage analysis of user cooperation. IEEE Comm. Lett., 17(8):1505-1508, 2013. URL:
  12. Lin Chen, Klaus Jansen, and Guochuan Zhang. On the optimality of exact and approximation algorithms for scheduling problems. J. Comput. Syst. Sci., 96:1-32, 2018. URL:
  13. Yookun Cho and Sartaj Sahni. Bounds for list schedules on uniform processors. SIAM J. Comput., 9(1):91-103, 1980. URL:
  14. Daniel Dadush, Arthur Léonard, Lars Rohwedder, and José Verschae. Optimizing low dimensional functions over the integers. In Integer Programming and Combinatorial Optimization, pages 115-126, 2023. URL:
  15. Friedrich Eisenbrand and Gennady Shmonin. Carathéodory bounds for integer cones. Oper. Res. Lett., 34(5):564-568, 2006. URL:
  16. Carlo Filippi and Giorgio Romanin-Jacur. Exact and approximate algorithms for high-multiplicity parallel machine scheduling. J. Sched., 12(5):529-541, 2009. URL:
  17. Teofilo Gonzalez, Oscar H. Ibarra, and Sartaj Sahni. Bounds for LPT schedules on uniform processors. SIAM J. Comput., 6(1):155-166, 1977. URL:
  18. Ronald L. Graham, Eugene L. Lawler, Jan K. Lenstra, and Alexander H. G. Rinnooy Kan. Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann. Discrete Math., 5:287-326, 1979. URL:
  19. Dmitry Gribanov, Ivan Shumilov, Dmitry Malyshev, and Panos Pardalos. On Δ-modular integer linear problems in the canonical form and equivalent problems. J. Glob. Optim., pages 1-61, 2022. URL:
  20. Martin Grötschel, László Lovász, and Alexander Schrijver. Geometric algorithms and combinatorial optimization, volume 2 of Algorithms and Combinatorics: Study and Research Texts. Springer Berlin, Heidelberg, 1988. URL:
  21. Christoph Haase and Georg Zetzsche. Presburger arithmetic with stars, rational subsets of graph groups, and nested zero tests. Proc. LICS 2019, pages 1-14, 2019. URL:
  22. Dorit S. Hochbaum and David B. Shmoys. A polynomial approximation scheme for scheduling on uniform processors: using the dual approximation approach. SIAM J. Comput., 17(3):539-551, 1988. URL:
  23. Klaus Jansen. An EPTAS for scheduling jobs on uniform processors: using an MILP relaxation with a constant number of integral variables. SIAM J. Discrete Math., 24(2):457-485, 2010. URL:
  24. Klaus Jansen. A fast approximation scheme for the multiple knapsack problem. In Proc. SOFSEM 2012, volume 7147 of Lecture Notes Comput. Sci., pages 313-324, 2012. URL:
  25. Klaus Jansen, Kim-Manuel Klein, and José Verschae. Closing the gap for makespan scheduling via sparsification techniques. In Proc. ICALP 2016, volume 55 of Leibniz Int. Proc. Informatics, pages Art. No. 72,13, 2016. URL:
  26. Klaus Jansen, Kim-Manuel Klein, and José Verschae. Closing the gap for makespan scheduling via sparsification techniques. Math. Oper. Res., 45(4):1371-1392, 2020. URL:
  27. Klaus Jansen and Marten Maack. An EPTAS for scheduling on unrelated machines of few different types. Algorithmica, 81(10):4134-4164, 2019. URL:
  28. Klaus Jansen and Christina Robenek. Scheduling jobs on identical and uniform processors revisited. In Proc. WAOA 2011, volume 7164 of Lecture Notes Comput. Sci., pages 109-122, 2012. URL:
  29. Ravi Kannan. Minkowski’s convex body theorem and integer programming. Math. Oper. Res., 12(3):415-440, 1987. URL:
  30. Viktor Kuncak and Martin Rinard. Towards efficient satisfiability checking for Boolean algebra with Presburger arithmetic. In Proc. CADE 2021, volume 4603 of Lecture Notes Comput. Sci., pages 215-230, 2007. URL:
  31. Hendrik W. Lenstra, Jr. Integer programming with a fixed number of variables. Math. Oper. Res., 8(4):538-548, 1983. URL:
  32. Jan Karel Lenstra, David B. Shmoys, and Éva Tardos. Approximation algorithms for scheduling unrelated parallel machines. Math. Programming, 46(3, (Ser. A)):259-271, 1990. URL:
  33. Pasin Manurangsi and Prasad Raghavendra. A birthday repetition theorem and complexity of approximating dense CSPs. Proc. ICALP 2017, 80:Art. No. 78,15, 2017. URL:
  34. Timm Oertel, Joseph Paat, and Robert Weismantel. Sparsity of integer solutions in the average case. Proc. IPCO 2019, 11480:341-353, 2019. URL:
  35. Ian Pratt-Hartmann. On the computational complexity of the numerically definite syllogistic and related logics. Bull. Symbolic Logic, 14(1):1-28, 2008. URL:
  36. Lars Rohwedder. Algorithms for Integer Programming and Allocation. phdthesis, Universität Kiel, 2019. URL:
  37. Thomas Rothvoss. Integer optimization and lattices, 2016. Lecture Notes. URL:
  38. Gerhard J. Woeginger. A comment on scheduling on uniform machines under chain-type precedence constraints. Oper. Res. Lett., 26(3):107-109, 2000. URL:
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