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Mathematical Programming

, Volume 158, Issue 1–2, pp 565–574 | Cite as

Extended formulations for sparsity matroids

  • Satoru Iwata
  • Naoyuki Kamiyama
  • Naoki Katoh
  • Shuji Kijima
  • Yoshio OkamotoEmail author
Short Communication Series A

Abstract

We show the existence of a polynomial-size extended formulation for the base polytope of a \((k,\ell )\)-sparsity matroid. For an undirected graph \(G=(V,E)\), the size of the formulation is \(O(|V|\cdot |E|)\) when \(k \ge \ell \) and \(O(|V|^2 |E|)\) when \(k \le \ell \). To this end, we employ the technique developed by Faenza et al. recently that uses a randomized communication protocol.

Mathematics Subject Classification

52B40 90C05 90C27 90C57 

Notes

Acknowledgments

The problem in this paper was partially discussed at ELC Workshop on Polyhedral Approaches: Extension Complexity and Pivoting Lower Bounds, held in Kyoto, Japan (June 2013). The authors thank the organizers and the participants of the workshop for stimulation to this work. Special thanks go to Hans Raj Tiwary for sharing his insights on randomized communication protocols. This work is supported by Grant-in-Aid for Scientific Research from Ministry of Education, Science and Culture, Japan and Japan Society for the Promotion of Science, and the ELC project (Grant-in-Aid for Scientific Research on Innovative Areas, MEXT Japan). The authors are also grateful to two anonymous referees for their constructive comments.

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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2015

Authors and Affiliations

  • Satoru Iwata
    • 1
  • Naoyuki Kamiyama
    • 2
  • Naoki Katoh
    • 3
  • Shuji Kijima
    • 4
  • Yoshio Okamoto
    • 5
    Email author
  1. 1.Department of Mathematical InformaticsUniversity of TokyoTokyoJapan
  2. 2.Institute of Mathematics for IndustryKyushu UniversityFukuokaJapan
  3. 3.Department of Architecture and Architectural EngineeringKyoto UniversityKyotoJapan
  4. 4.Department of InformaticsKyushu UniversityFukuokaJapan
  5. 5.Department of Communication Engineering and Informatics, Graduate School of Informatics and EngineeringThe University of Electro-CommunicationsChofuJapan

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