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Finding Diverse Solutions in Combinatorial Problems with a Distributive Lattice Structure

Authors Mark de Berg , Andrés López Martínez , Frits Spieksma

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  • Theory of computation → Design and analysis of algorithms
Keywords
  • Diversity
  • Lattice Theory
  • Submodular Function Minimization

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Abstract

We generalize the polynomial-time solvability of k-Diverse Minimum s-t Cuts (De Berg et al., ISAAC'23) to a wider class of combinatorial problems whose solution sets have a distributive lattice structure. We identify three structural conditions that, when met by a problem, ensure that a k-sized multiset of maximally-diverse solutions - measured by the sum of pairwise Hamming distances - can be found in polynomial time. We apply this framework to obtain polynomial-time algorithms for finding diverse minimum s-t cuts, diverse stable matchings, and diverse market-clearing price vectors. Moreover, we show that the framework extends to two other natural measures of diversity. Lastly, we present a simpler algorithmic framework for finding a largest set of pairwise disjoint solutions in problems that meet these structural conditions.

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Mark de Berg, Andrés López Martínez, and Frits Spieksma. Finding Diverse Solutions in Combinatorial Problems with a Distributive Lattice Structure. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ISAAC.2025.11

Author Details

Mark de Berg
  • Eindhoven University of Technology, The Netherlands
Andrés López Martínez
  • Eindhoven University of Technology, The Netherlands
Frits Spieksma
  • Eindhoven University of Technology, The Netherlands

Funding

This research was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 945045, and by the NWO Gravitation project NETWORKS under grant no. 024.002.003.

References

  1. Julien Baste, Lars Jaffke, Tomáš Masařík, Geevarghese Philip, and Günter Rote. Fpt algorithms for diverse collections of hitting sets. Algorithms, 12(12):254, 2019. URL: https://doi.org/10.3390/A12120254.
  2. Garrett Birkhoff. Rings of sets. Duke Mathematical Journal, 3(3):443-454, 1937. Google Scholar
  3. Charles Blair. The lattice structure of the set of stable matchings with multiple partners. Mathematics of operations research, 13(4):619-628, 1988. URL: https://doi.org/10.1287/MOOR.13.4.619.
  4. Mohammadreza Bolandnazar, Woonghee Tim Huh, S Thomas McCORMICK, and Kazuo Murota. A note on “order-based cost optimization in assemble-to-order systems”. University of Tokyo (February, Techical report, 2015. Google Scholar
  5. Paul Bonsma. Most balanced minimum cuts. Discrete Applied Mathematics, 158(4):261-276, 2010. URL: https://doi.org/10.1016/J.DAM.2009.09.010.
  6. Brian A Davey and Hilary A Priestley. Introduction to lattices and order. Cambridge university press, 2002. Google Scholar
  7. Mark de Berg, Andrés López Martínez, and Frits Spieksma. Finding diverse minimum st cuts. In 34th International Symposium on Algorithms and Computation, 2023. Google Scholar
  8. Mark de Berg, Andrés López Martínez, and Frits Spieksma. Finding diverse solutions in combinatorial problems with a distributive lattice structure, 2025. URL: https://doi.org/10.48550/arXiv.2504.02369.
  9. Karolina Drabik and Tomáš Masařík. Finding diverse solutions parameterized by cliquewidth. arXiv preprint, 2024. URL: https://arxiv.org/abs/2405.20931.
  10. David Easley, Jon Kleinberg, et al. Networks, crowds, and markets: Reasoning about a highly connected world, volume 1. Cambridge university press Cambridge, 2010. Google Scholar
  11. Fernando Escalante. Schnittverbände in graphen. In Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, volume 38, pages 199-220. Springer, 1972. Google Scholar
  12. Fedor V Fomin, Petr A Golovach, Lars Jaffke, Geevarghese Philip, and Danil Sagunov. Diverse pairs of matchings. Algorithmica, 86(6):2026-2040, 2024. URL: https://doi.org/10.1007/S00453-024-01214-7.
  13. Aadityan Ganesh, HV Vishwa Prakash, Prajakta Nimbhorkar, and Geevarghese Philip. Disjoint stable matchings in linear time. In Graph-Theoretic Concepts in Computer Science: 47th International Workshop, WG 2021, Warsaw, Poland, June 23-25, 2021, Revised Selected Papers 47, pages 94-105. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-86838-3_7.
  14. Rohith Reddy Gangam, Tung Mai, Nitya Raju, and Vijay V Vazirani. A structural and algorithmic study of stable matching lattices of "nearby" instances, with applications. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Schloss-Dagstuhl-Leibniz Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2022.19.
  15. Vijay K Garg. Introduction to lattice theory with computer science applications. John Wiley & Sons, 2015. Google Scholar
  16. Vijay K Garg. Applying predicate detection to the constrained optimization problems. arXiv preprint, 2018. URL: https://arxiv.org/abs/1812.10431.
  17. Vijay K Garg. Predicate detection to solve combinatorial optimization problems. In Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures, pages 235-245, 2020. URL: https://doi.org/10.1145/3350755.3400235.
  18. Vijay K Garg and Neeraj Mittal. On slicing a distributed computation. In Proceedings 21st International Conference on Distributed Computing Systems, pages 322-329. IEEE, 2001. URL: https://doi.org/10.1109/ICDSC.2001.918962.
  19. George Gratzer. Lattice theory: First concepts and distributive lattices. Courier Corporation, 2009. Google Scholar
  20. Martin Grötschel, László Lovász, and Alexander Schrijver. Geometric algorithms and combinatorial optimization, volume 2. Springer Science & Business Media, 2012. Google Scholar
  21. D. Gusfield and R.W. Irving. The Stable Marriage Problem: Structure and Algorithms. Foundations of computing. MIT Press, 1989. URL: https://books.google.nl/books?id=2TzhSQAACAAJ.
  22. R Halin. Lattices related to separation in graphs. In Finite and Infinite Combinatorics in Sets and Logic, pages 153-167. Springer, 1993. Google Scholar
  23. Tesshu Hanaka, Masashi Kiyomi, Yasuaki Kobayashi, Yusuke Kobayashi, Kazuhiro Kurita, and Yota Otachi. A framework to design approximation algorithms for finding diverse solutions in combinatorial problems. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 37, pages 3968-3976, 2023. URL: https://doi.org/10.1609/AAAI.V37I4.25511.
  24. Tesshu Hanaka, Yasuaki Kobayashi, Kazuhiro Kurita, and Yota Otachi. Finding diverse trees, paths, and more. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pages 3778-3786, 2021. URL: https://doi.org/10.1609/AAAI.V35I5.16495.
  25. Egbert Harzheim. Ordered sets, volume 7. Springer Science & Business Media, 2005. Google Scholar
  26. Yuni Iwamasa, Tomoki Matsuda, Shunya Morihira, and Hanna Sumita. A general framework for finding diverse solutions via network flow and its applications. arXiv preprint, 2025. URL: https://doi.org/10.48550/arXiv.2504.17633.
  27. Satoru Iwata, Lisa Fleischer, and Satoru Fujishige. A combinatorial strongly polynomial algorithm for minimizing submodular functions. Journal of the ACM (JACM), 48(4):761-777, 2001. URL: https://doi.org/10.1145/502090.502096.
  28. Donald Ervin Knuth. Stable marriage and its relation to other combinatorial problems: An introduction to the mathematical analysis of algorithms, volume 10. American Mathematical Soc., 1997. Google Scholar
  29. Soh Kumabe. Max-distance sparsification for diversification and clustering. arXiv preprint, 2024. URL: https://doi.org/10.48550/arXiv.2411.02845.
  30. George Markowsky. An overview of the poset of irreducibles. Combinatorial And Computational Mathematics, pages 162-177, 2001. Google Scholar
  31. Bernd Meyer. On the lattices of cutsets in finite graphs. European Journal of Combinatorics, 3(2):153-157, 1982. URL: https://doi.org/10.1016/S0195-6698(82)80028-0.
  32. Neeldhara Misra, Harshil Mittal, and Ashutosh Rai. On the parameterized complexity of diverse sat. In 35th International Symposium on Algorithms and Computation (ISAAC 2024), pages 50:1-50:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2024.50.
  33. Neeraj Mittal and Vijay K Garg. Computation slicing: Techniques and theory. In Distributed Computing: 15th International Conference, DISC 2001 Lisbon, Portugal, October 3-5, 2001 Proceedings 15, pages 78-92. Springer, 2001. URL: https://doi.org/10.1007/3-540-45414-4_6.
  34. Kazuo Murota. Discrete Convex Analysis. Society for Industrial and Applied Mathematics, 2003. URL: https://doi.org/10.1137/1.9780898718508.
  35. Jean-Claude Picard and Maurice Queyranne. On the structure of all minimum cuts in a network and applications. Math. Program., 22(1):121, December 1982. URL: https://doi.org/10.1007/BF01581031.
  36. Alexander Schrijver. A combinatorial algorithm minimizing submodular functions in strongly polynomial time. Journal of Combinatorial Theory, Series B, 80(2):346-355, 2000. URL: https://doi.org/10.1006/JCTB.2000.1989.
  37. Lloyd S Shapley and Martin Shubik. The assignment game I: The core. International Journal of game theory, 1(1):111-130, 1971. Google Scholar
  38. Yuto Shida, Giulia Punzi, Yasuaki Kobayashi, Takeaki Uno, and Hiroki Arimura. Finding diverse strings and longest common subsequences in a graph. In 35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024), pages 27:1-27:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.CPM.2024.27.
  39. Marilda Sotomayor. The lattice structure of the set of stable outcomes of the multiple partners assignment game. International Journal of Game Theory, 28:567-583, 1999. URL: https://doi.org/10.1007/S001820050126.
  40. Richard P Stanley. Enumerative combinatorics: Volume 1, 2011. Google Scholar
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