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.
This is a preview of subscription content, log in via an institution to check access.
References
Conforti, M., Kaibel, V., Walter, M., Weltge, S.: Subgraph polytopes and independence polytopes of count matroids. arXiv:1502.02817 [cs.DM] (2015)
Edmonds, J.: Matroids and the greedy algorithm. Math. Program. 1, 127–136 (1971)
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
Goemans, M.X.: Smallest compact formulation for the permutahedron. Math. Program. (2015). doi:10.1007/s10107-014-0757-1
Hakimi, S.L.: On the degrees of the vertices of a directed graphs. J. Frankl. Inst. 279, 290–308 (1965)
Hirahara, S., Imai, H.: On extended complexity of generalized transversal matroids. IEICE Technical Report, COMP2014-25 114, 1–4 (2014)
Imai, H.: Network-flow algorithms for lower-truncated transversal polymatroids. J. Oper. Res. Soc. Japan 26, 186–210 (1983)
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)
Laman, G.: On graphs and rigidity of plane skeletal structures. J. Eng. Math. 4, 331–340 (1970)
Martin, R.K.: Using separation algorithms to generate mixed integer model reformulations. Oper. Res. Lett. 10, 119–128 (1991)
Nash-Williams, C.S.J.A.: Edge-disjoint spanning trees of finite graphs. J. Lond. Math. Soc. s1–36, 445–450 (1961)
Rothvoss, T.: Some 0/1 polytopes need exponential size extended formulations. Math. Program., 142, 255–268 (2013)
Seymour, P.D.: Decomposition of regular matroids. J. Comb. Theo. Ser. B 28, 305–359 (1980)
Streinu, I., Theran, L.: Natural realizations of sparsity matroids. Ars Math. Contemp. 4, 141–151 (2011)
Tay, T.-S.: Rigidity of multi-graphs. I. Linking rigid bodies in \(n\)-space. J. Comb. Theo. Ser. B 36, 95–112 (1984)
Williams, J.C.: A linear-size zero-one programming model for the minimum spanning tree problem in planar graphs. Networks 39, 53–60 (2002)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Iwata, S., Kamiyama, N., Katoh, N. et al. Extended formulations for sparsity matroids. Math. Program. 158, 565–574 (2016). https://doi.org/10.1007/s10107-015-0936-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-015-0936-8