## Discrete Newton’s Algorithm for Parametric Submodular Function Minimization

##### Author(s)

Goemans, Michel X; Gupta, Swati; Jaillet, Patrick
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We consider the line search problem in a submodular polyhedron P (f) ⊆ ℝ n : Given an arbitrary a ∈ ℝ n and x 0 ∈ P (f), compute max{δ: x 0 + δa ∈ 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 Õ(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. Keywords:
Discrete Newton’s algorithm; Submodular functions; Line search; Ring families; Geometrically increasing sequence of sets; Fractional combinatorial optimization

##### Date issued

2017-05##### Department

Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science; Massachusetts Institute of Technology. Department of Mathematics; Sloan School of Management##### Journal

International Conference on Integer Programming and Combinatorial Optimization

##### Publisher

Springer-Verlag

##### Citation

Goemans, Michel X. et al. “Discrete Newton’s Algorithm for Parametric Submodular Function Minimization.” Lecture Notes in Computer Science (2017): 212–227 © 2017 Springer International Publishing AG

Version: Author's final manuscript

##### ISBN

978-3-319-59249-7

978-3-319-59250-3

##### ISSN

0302-9743

1611-3349