Extended formulations for sparsity matroids
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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 Classification52B40 90C05 90C27 90C57
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.
- 1.Conforti, M., Kaibel, V., Walter, M., Weltge, S.: Subgraph polytopes and independence polytopes of count matroids. arXiv:1502.02817 [cs.DM] (2015)
- 3.Faenza, Y., Fiorini, S., Grappe, R., Tiwary, H.R.: Extended formulations, nonnegative factorizations, and randomized communication protocols. Math. Program. (2015). doi: 10.1007/s10107-014-0755-3
- 4.Goemans, M.X.: Smallest compact formulation for the permutahedron. Math. Program. (2015). doi: 10.1007/s10107-014-0757-1
- 6.Hirahara, S., Imai, H.: On extended complexity of generalized transversal matroids. IEICE Technical Report, COMP2014-25 114, 1–4 (2014)Google Scholar
- 8.Kaibel, V., Lee, J., Walter, M., Weltge, S.: A quadratic upper bound on the extension complexities of the independence polytopes of regular matroids. arXiv:1504.03872 [cs.DM] (2015)