Skip to main content
SpringerLink
Log in

A Deterministic Parallel Reduction from Weighted Matroid Intersection Search to Decision

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

Given two matroids on the same ground set, the matroid intersection problem asks for a common base, i.e., a subset of the ground set that is a base in both the matroids. The weighted version of the problem asks for a common base with maximum weight. In the case of linearly representable matroids, the weighted version is known to have a randomized parallel (RNC) algorithm based on the isolation lemma, when the given weights are polynomially bounded (Narayanan et al. in SIAM J Comput 23(2): 387–397, 1994). Finding a deterministic parallel (NC) algorithm, even for the unweighted decision question, has been a long-standing open question. The above RNC algorithm can be viewed as a randomized reduction from weighted search to weighted decision, which works for arbitrary matroids. We derandomize this reduction, i.e., we give an NC algorithm for weighted matroid intersection search using oracle access to its decision version.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Algorithm 1
Algorithm 2

Similar content being viewed by others

Notes

  1. [30] does not describe the idea in terms of weights, but essentially they are assigning 0/1 weights to edges.

  2. Note that the independence oracle is known to be quite weak in the parallel model. Karp et al. [29] showed that with polynomially many parallel queries to the independence oracle in each round, it requires \(\Omega {(m/\log m)^{1/3}}\) rounds to find a base of a single matroid.

  3. In fact, they proved that the same lower bound on the number of rounds holds even if we allow \(2^{m^{1-\delta }}\) oracle calls for any constant \(\delta >0\)

  4. In case of independence oracle, their bound is \(O(m^{7/8})\) rounds.

  5. For a matroid M defined on the ground set E with \(e,f\in E\), \((M/e)-f = (M-f)/e\). The order of contractions/deletions does not matter.

References

  1. Agrawal, M., Gurjar, R., Korwar, A., Saxena, N.: Hitting-sets for ROABP and sum of set-multilinear circuits. SIAM J. Comput. 44(3), 669–697 (2015)

    Article  MathSciNet  Google Scholar 

  2. Anari, N., Vazirani, V.V: Matching is as easy as the decision problem, in the NC model. In: Proceedings of the 11th Innovations in Theoretical Computer Science Conference, ITCS 2020, January 12–14, 2020, Seattle, Washington, USA, volume 151 of LIPIcs, pp. 54:1–54:25. Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2020)

  3. Blikstad, J.: Sublinear-round parallel matroid intersection. In: Bojanczyk, M., Merelli, E., Woodruff, D.P. (eds.) 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4–8, 2022, Paris, France, volume 229 of LIPIcs, pp. 25:1–25:17. Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2022)

  4. Blikstad, J., van den Brand, J., Mukhopadhyay, S., Nanongkai, D.: Breaking the quadratic barrier for matroid intersection. In: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, pp. 421–432. Association for Computing Machinery, New York, NY, USA (2021)

  5. Chakrabarty, D., Chen, Y., Khanna, S.: A polynomial lower bound on the number of rounds for parallel submodular function minimization. In: 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7–10, 2022, pp. 37–48. IEEE (2021)

  6. Chari, S., Rohatgi, P., Srinivasan, A.: Randomness-optimal unique element isolation with applications to perfect matching and related problems. SIAM J. Comput. 24(5), 1036–1050 (1995)

    Article  MathSciNet  Google Scholar 

  7. Chakrabarty, D., Lee, Y.T., Sidford, A., Singla, S., Wong, S.C.-W.: Faster matroid intersection. In: Proceedings of the 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, pp. 1146–1168 (2019)

  8. Cunningham, W.H.: Improved bounds for matroid partition and intersection algorithms. SIAM J. Comput. 15(4), 948–957 (1986)

    Article  MathSciNet  Google Scholar 

  9. Dahlhaus, E., Karpinski, M.: Matching and multidimensional matching in chordal and strongly chordal graphs. Discrete Appl. Math. 84(1–3), 79–91 (1998)

    Article  MathSciNet  Google Scholar 

  10. Datta, S., Kulkarni, R., Roy, S.: Deterministically isolating a perfect matching in bipartite planar graphs. Theory Comput. Syst. 47, 737–757 (2010)

    Article  MathSciNet  Google Scholar 

  11. Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices. J. Res. Natl. Bureau Stand. Sect. B Math. Math. Phys. 69(125–130), 55–56 (1965)

    MathSciNet  Google Scholar 

  12. Edmonds, J.: Minimum partition of a matroid into independent subsets. J. Res. Natl. Bureau Stand. Sect. B Math. Math. Phys. 69, 67–72 (1965)

    Article  MathSciNet  Google Scholar 

  13. Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965)

    Article  MathSciNet  Google Scholar 

  14. Edmonds, J.: Matroid partition. Math. Decis. Sci. 11, 335–345 (1968)

    MathSciNet  Google Scholar 

  15. Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization- Eureka, You Shrink!, Papers Dedicated to Jack Edmonds, 5th International Workshop, Aussois, France, March 5–9, 2001, Revised Papers, volume 2570 of Lecture Notes in Computer Science, pp. 11–26. Springer (2001)

  16. Edmonds, J.: Matroid intersection. In: Johnson, E.L., Hammer, P.L., Korte, B.H. (eds.) Discrete Optimization I (Proceedings of the Advanced Research Institute on Discrete Optimization and Systems Applications of the Systems Science Panel of NATO and of the Discrete Optimization Symposium), vol. 4, pp. 39– 49. Elsevier (1979)

  17. Fenner, S.A., Gurjar, R., Thierauf, T.: Bipartite perfect matching is in quasi-NC. In: Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, pp. 754–763 (2016)

  18. Fredman, M.L., Komlós, J., Szemerédi, E.: Storing a sparse table with \(O(1)\) worst case access time. J. ACM 31(3), 538–544 (1984)

    Article  MathSciNet  Google Scholar 

  19. Frank, A.: A weighted matroid intersection algorithm. J. Algorithms 2(4), 328–336 (1981)

    Article  MathSciNet  Google Scholar 

  20. Goldwasser, S., Grossman, O.: Bipartite perfect matching in pseudo-deterministic NC. In: Chatzigiannakis, I., Indyk, P., Kuhn, F., Muscholl, A. (eds.) 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pp. 87:1–87:13. Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2017)

  21. Ghosh, S., Gurjar, R.: Matroid intersection: a pseudo-deterministic parallel reduction from search to weighted-decision. In: Proceedings of the Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2021, August 16–18, 2021, University of Washington, Seattle, Washington, USA (Virtual Conference), volume 207 of LIPIcs, pp. 41:1–41:16. Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2021)

  22. Ghosh, S., Gurjar, R., Raj, R.: A deterministic parallel reduction from weighted matroid intersection search to decision. In: Naor, J., Buchbinder, N. (eds.) Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference/Alexandria, VA, USA, January 9–12, 2022, pp. 1013–1035. SIAM (2022)

  23. Grigoriev, D., Karpinski, M.: The matching problem for bipartite graphs with polynomially bounded permanents is in NC (extended abstract). In: Proceedings of the 28th Annual Symposium on Foundations of Computer Science, FOCS 1987, pp. 166–172 (1987)

  24. Gurjar, R., Thierauf, T.: Linear matroid intersection is in quasi-NC. In: 49th Annual ACM Symposium on Theory of Computing, pp. 821–830 (2017)

  25. Harvey, N.J.A.: An algebraic algorithm for weighted linear matroid intersection. In: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’07, USA, pp. 444–453. Society for Industrial and Applied Mathematics (2007)

  26. Harvey, N.J.A.: An algebraic algorithm for weighted linear matroid intersection. Preliminary version appeared in SODA (2007). https://www.cs.ubc.ca/~nickhar/papers/WMI/WMI.pdf

  27. Klivans, A.R., Spielman, D.A.: Randomness efficient identity testing of multivariate polynomials. In: Proceedings on 33rd Annual ACM Symposium on Theory of Computing, July 6–8, 2001, Heraklion, Crete, Greece, pp. 216–223 (2001)

  28. Karp, R.M., Upfal, E., Wigderson, A.: Are search and decision programs computationally equivalent? In: Proceedings of the Seventeenth Annual ACM Symposium on Theory of Computing, STOC ’85, New York, NY, USA, pp. 464–475. Association for Computing Machinery (1985)

  29. Karp, R., Upfal, E., Wigderson, A.: The complexity of parallel computation on matroids. In: Proceedings of the 26th IEEE Annual Symposium on Foundations of Computer Science, FOCS 1985, pp. 541–550 (1985)

  30. Karp, R.M., Upfal, E., Wigderson, A.: Constructing a perfect matching is in random NC. Combinatorica 6(1), 35–48 (1986)

    Article  MathSciNet  Google Scholar 

  31. Karp, R.M., Upfal, E., Wigderson, A.: The complexity of parallel search. J. Comput. Syst. Sci. 36(2), 225–253 (1988)

    Article  MathSciNet  Google Scholar 

  32. Lawler, E.L.: Matroid intersection algorithms. Math. Program. 9(1), 31–56 (1975)

    Article  MathSciNet  Google Scholar 

  33. Lovász, L.: On determinants, matchings, and random algorithms. In: Budach, L. (ed.) Fundamentals of Computation Theory, FCT 1979, Proceedings of the Conference on Algebraic, Arthmetic, and Categorial Methods in Computation Theory, Berlin/Wendisch-Rietz, Germany, September 17–21, 1979, pp. 565–574. Akademie-Verlag, Berlin (1979)

  34. Mulmuley, K., Vazirani, U.V., Vazirani, V.V.: Matching is as easy as matrix inversion. Combinatorica 7, 105–113 (1987)

    Article  MathSciNet  Google Scholar 

  35. Nguyen, H.L.: A note on Cunningham’s algorithm for matroid intersection. arXiv e-prints arXiv:1904.04129, April (2019)

  36. Narayanan, H., Saran, H., Vazirani, V.V.: Randomized parallel algorithms for matroid union and intersection, with applications to arboresences and edge-disjoint spanning trees. SIAM J. Comput. 23(2), 387–397 (1994)

    Article  MathSciNet  Google Scholar 

  37. Oxley, J.G.: Matroid Theory (Oxford Graduate Texts in Mathematics). Oxford University Press Inc., New York (2006)

    Google Scholar 

  38. Rabin, M.O., Vazirani, V.V.: Maximum matchings in general graphs through randomization. J. Algorithms 10(4), 557–567 (1989)

    Article  MathSciNet  Google Scholar 

  39. Schrijver, A.: Combinatorial optimization: polyhedra and efficiency. Vol. B., matroids, trees, stable sets. Chapters 39–69. In: Algorithms and Combinatorics. Springer-Verlag, Berlin, Heidelberg, New York (2003)

  40. Svensson, O., Tarnawski, J.: The matching problem in general graphs is in quasi-NC. In: 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15–17, 2017, pp. 696–707 (2017)

  41. Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43(3), 441–466 (1991)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The second author appreciates funding support from the SERB MATRICS grant. We thank the anonymous reviewers for their suggestions and comments on earlier versions of the manuscript.

Author information

Authors and Affiliations

  1. Computer Science, Chennai Mathematical Institute, Chennai, India

    Sumanta Ghosh

  2. Department of Computer Science, IIT Bombay, Mumbai, India

    Sumanta Ghosh

  3. Department of Computer Science and Engineering, IIT Bombay, Mumbai, India

    Rohit Gurjar & Roshan Raj

Authors
  1. Sumanta Ghosh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rohit Gurjar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Roshan Raj
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally.

Corresponding author

Correspondence to Sumanta Ghosh.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

A preliminary version of this paper is published in the Proceedings of SODA 2022, and it is available at the following link: https://epubs.siam.org/doi/epdf/10.1137/1.9781611977073.44.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ghosh, S., Gurjar, R. & Raj, R. A Deterministic Parallel Reduction from Weighted Matroid Intersection Search to Decision. Algorithmica 86, 1057–1079 (2024). https://doi.org/10.1007/s00453-023-01184-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-023-01184-2

Keywords

Search

Navigation