Discrete Newton's Algorithm for Parametric Submodular Function Minimization

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.