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Tight Algorithm and Hardness for Submodular Linear Ordering

Authors Evan Abboud , Roy Schwartz

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Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Submodular optimization and polymatroids
Keywords
  • Submodular optimization
  • approximation algorithms
  • hardness of approximation
  • linear ordering
  • combinatorial optimization

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Abstract

We consider the Minimum Linear Ordering Problem: given a ground set N of cardinality n and a non-negative set function f: 2^N → ℝ_{≥0}, the goal is to find an ordering π of N that minimizes the sum of the values of f over all prefixes of π. This problem has been studied for various classes of set functions, and the case of a submodular f is of special interest, as it captures classic problems including Minimum Linear Arrangement and Minimum Containing Interval Graph. In this work, we resolve the approximability of the Minimum Linear Ordering Problem for a general submodular f by establishing matching upper and lower bounds and present: (1) a polynomial-time algorithm achieving an O(√{n/ln n})-approximation; and (2) a matching information-theoretic hardness result, showing that no algorithm evaluating f a polynomial number of times can achieve an o(√{n/ln n})-approximation. Previously, the best known hardness of approximation was 2, and an O(√{n/ln n})-approximation was known only for the special case where f is both submodular and symmetric.

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Evan Abboud and Roy Schwartz. Tight Algorithm and Hardness for Submodular Linear Ordering. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.4

Author Details

Evan Abboud
  • Technion - Israel Institute of Technology, Haifa, Israel
Roy Schwartz
  • Technion - Israel Institute of Technology, Haifa, Israel

Funding

Evan Abboud and Roy Schwartz received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement no. 852870-ERC-SUBMODULAR and ISF grant 3264/25.

Acknowledgements

The authors would like to thank Yuval Filmus for stimulating discussions.

References

  1. Sanjeev Arora, Satish Rao, and Umesh Vazirani. Expander flows, geometric embeddings and graph partitioning. Journal of the ACM (JACM), 56(2):1-37, 2009. URL: https://doi.org/10.1145/1502793.1502794.
  2. Yossi Azar, Iftah Gamzu, and Ran Roth. Submodular max-sat. In European Symposium on Algorithms, pages 323-334. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-23719-5_28.
  3. Nikhil Bansal, Jatin Batra, Majid Farhadi, and Prasad Tetali. Improved approximations for min sum vertex cover and generalized min sum set cover. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 998-1005. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.62.
  4. Rémi Bardenet and Odalric-Ambrym Maillard. Concentration inequalities for sampling without replacement. arXiv preprint, 2013. URL: https://arxiv.org/abs/1309.4029.
  5. Uri Barenholz, Uriel Feige, David Peleg, et al. Improved approximation for min-sum vertex cover. Technical Report MCS06-07, Computer Science and Applied Mathematics, Weizmann Institute of Science, 2006. Google Scholar
  6. Sandeep N. Bhatt and Frank Thomson Leighton. A framework for solving VLSI graph layout problems. J. Comput. Syst. Sci., 28(2):300-343, 1984. URL: https://doi.org/10.1016/0022-0000(84)90071-0.
  7. Moses Charikar, Mohammad Taghi Hajiaghayi, Howard Karloff, and Satish Rao. 𝓁₂² spreading metrics for vertex ordering problems. Algorithmica, 56(4):577-604, 2010. URL: https://doi.org/10.1007/S00453-008-9191-1.
  8. Josep Díaz, AM Gibbons, MS Paterson, and Jacobo Toran. The minsumcut problem. In Workshop on Algorithms and Data Structures, pages 65-79. Springer, 1991. URL: https://doi.org/10.1007/BFB0028251.
  9. Shahar Dobzinski and Jan Vondrák. From query complexity to computational complexity. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 1107-1116, 2012. URL: https://doi.org/10.1145/2213977.2214076.
  10. Alina Ene, Jan Vondrák, and Yi Wu. Local distribution and the symmetry gap: Approximability of multiway partitioning problems. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 306-325. SIAM, 2013. URL: https://doi.org/10.1137/1.9781611973105.23.
  11. Guy Even, Joseph Seffi Naor, Satish Rao, and Baruch Schieber. Divide-and-conquer approximation algorithms via spreading metrics. Journal of the ACM (JACM), 47(4):585-616, 2000. URL: https://doi.org/10.1145/347476.347478.
  12. Majid Farhadi, Swati Gupta, Shengding Sun, Prasad Tetali, and Michael C Wigal. Hardness and approximation of submodular minimum linear ordering problems. Mathematical Programming, 208(1):277-318, 2024. URL: https://doi.org/10.1007/S10107-023-02038-Z.
  13. Uriel Feige, MohammadTaghi Hajiaghayi, and James R. Lee. Improved approximation algorithms for minimum weight vertex separators. SIAM Journal on Computing, 38(2):629-657, 2008. URL: https://doi.org/10.1137/05064299X.
  14. Uriel Feige and James R Lee. An improved approximation ratio for the minimum linear arrangement problem. Information Processing Letters, 101(1):26-29, 2007. URL: https://doi.org/10.1016/J.IPL.2006.07.009.
  15. Uriel Feige, László Lovász, and Prasad Tetali. Approximating min sum set cover. Algorithmica, 40(4):219-234, 2004. URL: https://doi.org/10.1007/S00453-004-1110-5.
  16. Uriel Feige, Vahab S Mirrokni, and Jan Vondrák. Maximizing non-monotone submodular functions. SIAM Journal on Computing, 40(4):1133-1153, 2011. URL: https://doi.org/10.1137/090779346.
  17. Yuval Filmus, Roy Schwartz, and Alexander V Smal. Separating coverage and submodular: Maximization subject to a cardinality constraint. In International Conference on Integer Programming and Combinatorial Optimization, pages 242-255. Springer, 2025. URL: https://doi.org/10.1007/978-3-031-93112-3_18.
  18. Shayan Oveis Gharan and Jan Vondrák. Submodular maximization by simulated annealing. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms, pages 1098-1116. SIAM, 2011. URL: https://doi.org/10.1137/1.9781611973082.83.
  19. Gagan Goel, Chinmay Karande, Pushkar Tripathi, and Lei Wang. Approximability of combinatorial problems with multi-agent submodular cost functions. In 2009 50th Annual IEEE Symposium on Foundations of Computer Science, pages 755-764. IEEE Computer Society, 2009. URL: https://doi.org/10.1109/FOCS.2009.81.
  20. Michel X Goemans, Nicholas JA Harvey, Satoru Iwata, and Vahab Mirrokni. Approximating submodular functions everywhere. In Proceedings of the twentieth annual ACM-SIAM symposium on Discrete algorithms, pages 535-544. SIAM, 2009. URL: https://doi.org/10.1137/1.9781611973068.59.
  21. Refael Hassin and Asaf Levin. An approximation algorithm for the minimum latency set cover problem. In European Symposium on Algorithms, pages 726-733. Springer, 2005. URL: https://doi.org/10.1007/11561071_64.
  22. Wassily Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American statistical association, 58(301):13-30, 1963. URL: https://doi.org/10.1080/01621459.1963.10500830.
  23. Satoru Iwata and Kiyohito Nagano. Submodular function minimization under covering constraints. In 2009 50th annual IEEE symposium on foundations of computer science, pages 671-680. IEEE, 2009. URL: https://doi.org/10.1109/FOCS.2009.31.
  24. Satoru Iwata, Prasad Tetali, and Pushkar Tripathi. Approximating minimum linear ordering problems. In International Workshop on Approximation Algorithms for Combinatorial Optimization, pages 206-217. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-32512-0_18.
  25. Dor Katzelnick and Roy Schwartz. A simple algorithm for submodular minimum linear ordering. In Symposium on Simplicity in Algorithms (SOSA), pages 28-35. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977585.CH3.
  26. Tom Leighton and Satish Rao. Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. Journal of the ACM (JACM), 46(6):787-832, 1999. URL: https://doi.org/10.1145/331524.331526.
  27. Yixun Lin and Jinjiang Yuan. Profile minimization problem for matrices and graphs. Acta Mathematicae Applicatae Sinica, 10:107-112, 1994. URL: https://doi.org/10.1007/BF02006264.
  28. László Lovász. Submodular functions and convexity. In Mathematical Programming The State of the Art: Bonn 1982, pages 235-257. Springer, 1983. URL: https://doi.org/10.1007/978-3-642-68874-4_10.
  29. Vahab Mirrokni, Michael Schapira, and Jan Vondrák. Tight information-theoretic lower bounds for welfare maximization in combinatorial auctions. In Proceedings of the 9th ACM conference on Electronic commerce, pages 70-77, 2008. URL: https://doi.org/10.1145/1386790.1386805.
  30. G. L. Nemhauser and M. L. Fisher. Best algorithms for approximating the maximum of a submodular set function. Mathematics of Operations Research, 3(3):177-188, 1978. URL: https://doi.org/10.1287/MOOR.3.3.177.
  31. G. Ramalingam and C.Pandu Rangan. A unified approach to domination problems on interval graphs. Information Processing Letters, 27(5):271-274, 1988. URL: https://doi.org/10.1016/0020-0190(88)90091-9.
  32. Satish Rao and Andréa W Richa. New approximation techniques for some linear ordering problems. SIAM Journal on Computing, 34(2):388-404, 2005. URL: https://doi.org/10.1137/S0097539702413197.
  33. Ramamurthy Ravi, Ajit Agrawal, and Philip Klein. Ordering problems approximated: single-processor scheduling and interval graph completion. In International Colloquium on Automata, Languages, and Programming, pages 751-762. Springer, 1991. URL: https://doi.org/10.1007/3-540-54233-7_180.
  34. Richard Santiago and F Bruce Shepherd. Multivariate submodular optimization. In International Conference on Machine Learning, pages 5599-5609. PMLR, 2019. URL: http://proceedings.mlr.press/v97/santiago19a.html.
  35. Paul D. Seymour. Packing directed circuits fractionally. Combinatorica, 15(2):281-288, 1995. URL: https://doi.org/10.1007/BF01200760.
  36. Aleksa Stanković. Some Results on Approximability of Minimum Sum Vertex Cover. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022), volume 245, pages 50:1-50:16, 2022. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.50.
  37. Zoya Svitkina and Lisa Fleischer. Submodular approximation: Sampling-based algorithms and lower bounds. SIAM Journal on Computing, 40(6):1715-1737, 2011. URL: https://doi.org/10.1137/100783352.
  38. Jan Vondrák. Symmetry and approximability of submodular maximization problems. SIAM Journal on Computing, 42(1):265-304, 2013. URL: https://doi.org/10.1137/110832318.
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