Abstract
We consider the line search problem in a submodular polyhedron \(P(f)\subseteq {\mathbb {R}}^n\): Given an arbitrary \(a\in {\mathbb {R}}^n\) and \(x_0\in P(f)\), compute \(\max \{\delta : x_0+\delta a\in P(f)\}\). The use of the discrete Newton’s algorithm for this line search problem is very natural, but no strongly polynomial bound on its number of iterations was known (Iwata 2008). We solve this open problem by providing a quadratic bound of \(n^2 + O(n \log ^2 n)\) on its number of iterations. Our result considerably improves upon the only other known strongly polynomial time algorithm, which is based on Megiddo’s parametric search framework and which requires \({\tilde{O}}(n^8)\) submodular function minimizations (Nagano 2007). As a by-product of our study, we prove (tight) bounds on the length of chains of ring families and geometrically increasing sequences of sets, which might be of independent interest.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Freund, R.M., Grigas, P., Mazumder, R.: An extended Frank-Wolfe method with “In-Face" directions, its application to low-rank matrix completion (2015). arXiv:1511.02204
Håstad, J.: On the size of weights for threshold gates. SIAM J. Discrete Math. 7(3), 484–492 (1994)
Iwata, S.: Submodular function minimization. Math. Program. 112(1), 45–64 (2008)
Iwata, S., Murota, K., Shigeno, M.: A fast parametric submodular intersection algorithm for strong map sequences. Math. Oper. Res. 22(4), 803–813 (1997)
Iwata, S., Orlin, J.B.: A simple combinatorial algorithm for submodular function minimization. In: Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1230–1237. Society for Industrial and Applied Mathematics (2009)
Lee, Y.T., Sidford, A., Wong, S.C.: A faster cutting plane method and its implications for combinatorial and convex optimization. In: Foundations of Computer Science (FOCS), pp. 1049–1065. IEEE (2015)
McCormick, S.T., Ervolina, T.R.: Computing maximum mean cuts. Discrete Appl. Math. 52(1), 53–70 (1994)
Mulmuley, K.: Lower bounds in a parallel model without bit operations. SIAM J. Comput. 28(4), 1460–1509 (1999)
Nagano, K.: A strongly polynomial algorithm for line search in submodular polyhedra. Discrete Optim. 4(3), 349–359 (2007)
Radzik, T.: Fractional combinatorial optimization. In: Du, D.Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization, pp. 429–478. Springer, Heidelberg (1998)
Topkis, D.M.: Minimizing a submodular function on a lattice. Oper. Res. 26(2), 305–321 (1978)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Goemans, M.X., Gupta, S., Jaillet, P. (2017). Discrete Newton’s Algorithm for Parametric Submodular Function Minimization. In: Eisenbrand, F., Koenemann, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2017. Lecture Notes in Computer Science(), vol 10328. Springer, Cham. https://doi.org/10.1007/978-3-319-59250-3_18
Download citation
DOI: https://doi.org/10.1007/978-3-319-59250-3_18
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-59249-7
Online ISBN: 978-3-319-59250-3
eBook Packages: Computer ScienceComputer Science (R0)