Skip to main content
SpringerLink
Log in

Critical Rows of Almost-Factorable Matrices

  • Conference paper
  • First Online:
Combinatorial Optimization and Applications (COCOA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11949))

  • 795 Accesses

Abstract

An almost-factorable matrix is a matrix that has a k-rank nonnegative factorization after deleting a single row. We call such a row a critical row. In graph optimization, a critical edge is an edge which can be deleted to reduce a graph measure, such as the size of the minimum vertex cover. We prove a reduction between critical rows in matrices and critical edges in graphs. Additionally, we describe and experimentally test an algorithm to identify the critical row of a matrix.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Balaji, S., Swaminathan, V., Kannan, K.: Optimization of unweighted minimum vertex cover. World Acad. Sci. Eng. Technol. 43, 716–729 (2010)

    Google Scholar 

  2. Brunet, J.P., Tamayo, P., Golub, T.R., Mesirov, J.P.: Metagenes and molecular pattern discovery using matrix factorization. Proc. Nat. Acad. Sci. 101(12), 4164–4169 (2004)

    Article  Google Scholar 

  3. Clarkson, K.L.: A modification of the greedy algorithm for vertex cover. Inf. Process. Lett. 16(1), 23–25 (1983)

    Article  MathSciNet  Google Scholar 

  4. Computational Intelligence Group, U.o.B.C. http://www.ehu.eus/ccwintco/index.php/hyperspectral_remote_sensing_Scenes

  5. Erdos, P., Gallai, T.: On the minimal number of vertices of a graph representing the edges of a graph (1961)

    Google Scholar 

  6. Gillis, N.: The why and how of nonnegative matrix factorization. Regul. Optim. Kernels Support. Vector Mach. 12(257), 257–291 (2014)

    Google Scholar 

  7. Jakoby, A., Goswami, N.K., List, E., Lucks, S.: Critical graphs for minimum vertex cover. CoRR abs/1705.04111 (2017)

    Google Scholar 

  8. Jiang, T., Ravikumar, B.: Minimal NFA problems are hard. SIAM J. Comput. 22(6), 1117–1141 (1993)

    Article  MathSciNet  Google Scholar 

  9. Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of computer computations. The IBM Research Symposia Series, pp. 85–103. Springer, Boston (1972). https://doi.org/10.1007/978-1-4684-2001-2_9

    Chapter  Google Scholar 

  10. Khan, I., Khan, S.: Experimental comparison of five approximation algorithms for minimum vertex cover. Int. J. u-and e-Serv. 7(6), 69–84 (2014)

    Google Scholar 

  11. Lee, D.D., Seung, H.S.: Algorithms for non-negative matrix factorization. In: Advances in Neural Information Processing Systems, pp. 556–562 (2001)

    Google Scholar 

  12. Miettinen, P., Mielikäinen, T., Gionis, A., Das, G., Mannila, H.: The discrete basis problem. In: Fürnkranz, J., Scheffer, T., Spiliopoulou, M. (eds.) PKDD 2006. LNCS (LNAI), vol. 4213, pp. 335–346. Springer, Heidelberg (2006). https://doi.org/10.1007/11871637_33

    Chapter  Google Scholar 

  13. Moitra, A.: An almost optimal algorithm for computing nonnegative rank. SIAM J. Comput. 45(1), 156–173 (2016)

    Article  MathSciNet  Google Scholar 

  14. Thomas, L.: Rank factorization of nonnegative matrices. SIAM Rev. 16(3), 393–394 (1974). https://doi.org/10.1137/1016064

    Article  Google Scholar 

  15. Vavasis, S.A.: On the complexity of nonnegative matrix factorization. SIAM J. Optim. 20(3), 1364–1377 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Pennsylvania, Philadelphia, PA, USA

    Peter Ballen

Authors
  1. Peter Ballen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Ballen .

Editor information

Editors and Affiliations

  1. Georgia State University, Atlanta, GA, USA

    Yingshu Li

  2. Florida Atlantic University, Boca Raton, FL, USA

    Mihaela Cardei

  3. Kennesaw State University, Marietta, GA, USA

    Yan Huang

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Ballen, P. (2019). Critical Rows of Almost-Factorable Matrices. In: Li, Y., Cardei, M., Huang, Y. (eds) Combinatorial Optimization and Applications. COCOA 2019. Lecture Notes in Computer Science(), vol 11949. Springer, Cham. https://doi.org/10.1007/978-3-030-36412-0_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-36412-0_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-36411-3

  • Online ISBN: 978-3-030-36412-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation