Discrete Newton's Algorithm for Parametric Submodular Function Minimization

We consider the line search problem in a submodular polytope P (f) ⊆ R: Given an arbitrary a ∈ R and x0 ∈ P (f), compute max{δ : x0 + 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 + O(n log 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) 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.