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Polynomial-Delay Enumeration of Large Maximal Common Independent Sets in Two Matroids

Authors Yasuaki Kobayashi , Kazuhiro Kurita , Kunihiro Wasa

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Author Details

Yasuaki Kobayashi
  • Hokkaido University, Sapporo, Japan
Kazuhiro Kurita
  • Nagoya University, Nagoya, Japan
Kunihiro Wasa
  • Hosei University, Tokyo, Japan

Funding

  • Kobayashi, Yasuaki: JSPS KAKENHI Grant Numbers JP20H00595 and JP23H03344
  • Kurita, Kazuhiro: JP21K17812, JP22H03549, JST CREST Grant Number JPMJCR18K3, and JST ACT-X Grant Number JPMJAX2105
  • Wasa, Kunihiro: JSPS KAKENHI Grant Numbers JP22K17849 and JP22H03549, and JST CREST Grant Number JPMJCR18K3

Cite As Get BibTex

Yasuaki Kobayashi, Kazuhiro Kurita, and Kunihiro Wasa. Polynomial-Delay Enumeration of Large Maximal Common Independent Sets in Two Matroids. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 58:1-58:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.MFCS.2023.58

Abstract

Finding a maximum cardinality common independent set in two matroids (also known as Matroid Intersection) is a classical combinatorial optimization problem, which generalizes several well-known problems, such as finding a maximum bipartite matching, a maximum colorful forest, and an arborescence in directed graphs. Enumerating all maximal common independent sets in two (or more) matroids is a classical enumeration problem. In this paper, we address an "intersection" of these problems: Given two matroids and a threshold τ, the goal is to enumerate all maximal common independent sets in the matroids with cardinality at least τ. We show that this problem can be solved in polynomial delay and polynomial space. We also discuss how to enumerate all maximal common independent sets of two matroids in non-increasing order of their cardinalities.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Enumeration
Keywords
  • Polynomial-delay enumeration
  • Ranked Enumeration
  • Matroid intersection
  • Reverse search

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References

  1. D. Avis and K. Fukuda. Reverse search for enumeration. Discrete Applied Mathematics, 65(1):21-46, 1996. URL: https://doi.org/10.1016/0166-218X(95)00026-N.
  2. Etienne Birmelé, Rui Ferreira, Roberto Grossi, Andrea Marino, Nadia Pisanti, Romeo Rizzi, and Gustavo Sacomoto. Optimal listing of cycles and st-paths in undirected graphs. In Proceedings of the 2013 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1884-1896. SIAM, 2013. URL: https://doi.org/10.1137/1.9781611973105.134.
  3. Endre Boros, Khaled Elbassioni, Vladimir Gurvich, and Leonid Khachiyan. Matroid intersections, polymatroid inequalities, and related problems. In Krzysztof Diks and Wojciech Rytter, editors, Mathematical Foundations of Computer Science 2002, pages 143-154, Berlin, Heidelberg, 2002. Springer Berlin Heidelberg. Google Scholar
  4. Florent Capelli and Yann Strozecki. Geometric Amortization of Enumeration Algorithms. In Petra Berenbrink, Patricia Bouyer, Anuj Dawar, and Mamadou Moustapha Kanté, editors, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023), volume 254 of Leibniz International Proceedings in Informatics (LIPIcs), pages 18:1-18:22, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.STACS.2023.18.
  5. Katrin Casel, Henning Fernau, Mehdi Khosravian Ghadikolaei, Jérôme Monnot, and Florian Sikora. Extension of some edge graph problems: Standard and parameterized complexity. In Proc. FCT 2019, volume 11651 of Lecture Notes in Computer Science, pages 185-200. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-25027-0_13.
  6. Alessio Conte, Roberto Grossi, Andrea Marino, Takeaki Uno, and Luca Versari. Listing maximal independent sets with minimal space and bounded delay. In Gabriele Fici, Marinella Sciortino, and Rossano Venturini, editors, String Processing and Information Retrieval - 24th International Symposium, SPIRE 2017, Palermo, Italy, September 26-29, 2017, Proceedings, volume 10508 of Lecture Notes in Computer Science, pages 144-160. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-67428-5_13.
  7. William H. Cunningham. Improved bounds for matroid partition and intersection algorithms. SIAM Journal on Computing, 15(4):948-957, 1986. URL: https://doi.org/10.1137/0215066.
  8. Shaleen Deep and Paraschos Koutris. Ranked Enumeration of Conjunctive Query Results. In Ke Yi and Zhewei Wei, editors, 24th International Conference on Database Theory (ICDT 2021), volume 186 of Leibniz International Proceedings in Informatics (LIPIcs), pages 5:1-5:19, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICDT.2021.5.
  9. Jack Edmonds. Paths, trees, and flowers. Can. J. Math., 17:449-467, 1965. URL: https://doi.org/10.4153/CJM-1965-045-4.
  10. Jack Edmonds. Submodular functions, matroids, and certain polyhedra. In R. Guy, H. Hanani, N. Sauer, and J. Schönheim, editors, Combinatorial Structures and Their Applications, (Proceedings, Calgary International Conference on Combinatorial Structures and Their Applications), pages 69-87, Calgary, Alberta, 1970. Google Scholar
  11. David Eppstein. k-Best Enumeration. In Encyclopedia of Algorithms, pages 1003-1006. Springer, New York, NY, 2016. URL: https://doi.org/10.1007/978-1-4939-2864-4_733.
  12. Komei Fukuda and Makoto Namiki. Finding all common bases in two matroids. Discrete Applied Mathematics, 56(2):231-243, 1995. Fifth Franco-Japanese Days. URL: https://doi.org/10.1016/0166-218X(94)00088-U.
  13. David S. Johnson, Mihalis Yannakakis, and Christos H. Papadimitriou. On generating all maximal independent sets. Inform. Process. Lett., 27(3):119-123, 1988. URL: https://doi.org/10.1016/0020-0190(88)90065-8.
  14. Mamadou Moustapha Kanté, Vincent Limouzy, Arnaud Mary, Lhouari Nourine, and Takeaki Uno. Polynomial Delay Algorithm for Listing Minimal Edge Dominating Sets in Graphs. In Proc. of WADS 2015, pages 446-457, 2015. URL: https://doi.org/10.1007/978-3-319-21840-3_37.
  15. L. Khachiyan, E. Boros, K. Borys, K. Elbassioni, V. Gurvich, and K. Makino. Enumerating spanning and connected subsets in graphs and matroids. In Yossi Azar and Thomas Erlebach, editors, Algorithms - ESA 2006, pages 444-455, Berlin, Heidelberg, 2006. Springer Berlin Heidelberg. Google Scholar
  16. Leonid Khachiyan, Endre Boros, Konrad Borys, Khaled M. Elbassioni, Vladimir Gurvich, and Kazuhisa Makino. Generating Cut Conjunctions in Graphs and Related Problems. Algorithmica, 51(3):239-263, 2008. URL: https://doi.org/10.1007/s00453-007-9111-9.
  17. Leonid G. Khachiyan, Endre Boros, Khaled M. Elbassioni, Vladimir Gurvich, and Kazuhisa Makino. On the complexity of some enumeration problems for matroids. SIAM J. Discret. Math., 19(4):966-984, 2005. URL: https://doi.org/10.1137/S0895480103428338.
  18. Yasuaki Kobayashi, Kazuhiro Kurita, and Kunihiro Wasa. Efficient constant-factor approximate enumeration of minimal subsets for monotone properties with weight constraints. CoRR, abs/2009.08830, 2020. URL: https://arxiv.org/abs/2009.08830.
  19. Yasuaki Kobayashi, Kazuhiro Kurita, and Kunihiro Wasa. Polynomial-delay and polynomial-space enumeration of large maximal matchings. In Michael A. Bekos and Michael Kaufmann, editors, Graph-Theoretic Concepts in Computer Science, pages 342-355, Cham, 2022. Springer International Publishing. Google Scholar
  20. Tuukka Korhonen. Listing small minimal separators of a graph. CoRR, abs/2012.09153, 2020. URL: https://arxiv.org/abs/2012.09153.
  21. Kazuhiro Kurita and Yasuaki Kobayashi. Efficient enumerations for minimal multicuts and multiway cuts. In Proc. MFCS 2020, pages 60:1-60:14, 2020. URL: https://doi.org/10.4230/LIPIcs.MFCS.2020.60.
  22. E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan. Generating All Maximal Independent Sets: NP-Hardness and Polynomial-Time Algorithms. SIAM Journal on Computing, 9(3):558-565, 1980. Google Scholar
  23. Eugene L Lawler. Matroid intersection algorithms. Mathematical Programming, 9(1):31-56, 1975. URL: https://doi.org/10.1007/BF01681329.
  24. Kazuhisa Makino and Takeaki Uno. New algorithms for enumerating all maximal cliques. In Proc. SWAT 2004, volume 3111 of Lecture Notes in Computer Science, pages 260-272. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-27810-8_23.
  25. James Oxley. Matroid theory, volume 3. Oxford University Press, USA, 2006. Google Scholar
  26. Noam Ravid, Dori Medini, and Benny Kimelfeld. Ranked Enumeration of Minimal Triangulations. In Proc. of PODS 2019, pages 74-88, 2019. URL: https://doi.org/10.1145/3294052.3319678.
  27. R. C. Read and R. E. Tarjan. Bounds on backtrack algorithms for listing cycles, paths, and spanning trees. Networks, 5(3):237-252, 1975. URL: https://doi.org/10.1002/net.1975.5.3.237.
  28. Alexander Schrijver. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer Science & Business Media, 2003. Google Scholar
  29. Shuji Tsukiyama, Mikio Ide, Hiromu Ariyoshi, and Isao Shirakawa. A New Algorithm for Generating All the Maximal Independent Sets. SIAM J. Comput., 6(3):505-517, 1977. URL: https://doi.org/10.1137/0206036.
  30. Takeaki Uno. Algorithms for enumerating all perfect, maximum and maximal matchings in bipartite graphs. In Hon Wai Leong, Hiroshi Imai, and Sanjay Jain, editors, Algorithms and Computation, pages 92-101, Berlin, Heidelberg, 1997. Springer Berlin Heidelberg. Google Scholar
  31. Takeaki Uno. A fast algorithm for enumerating bipartite perfect matchings. In Peter Eades and Tadao Takaoka, editors, Algorithms and Computation, pages 367-379, Berlin, Heidelberg, 2001. Springer Berlin Heidelberg. Google Scholar
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