Faster Parametric Submodular Function Minimization by Exploiting Duality

Let $f:2^{E} \rightarrow \mathbb{Z}_+$ be a submodular function on a ground set $E = [n]$, and let $P(f)$ denote its extended polymatroid. Given a direction $d \in \mathbb{Z}^n$ with at least one positive entry, the line search problem is to find the largest scalar $\lambda$ such that $\lambda d \in P(f)$. The best known strongly polynomial-time algorithm for this problem is based on the discrete Newton's method and requires $\tilde{O}(n^2 \log n)\cdot$ SFM time, where SFM is the time for exact submodular function minimization under the value oracle model. In this work, we study the first weakly polynomial-time algorithms for this problem. We reduce the number of calls to the exact submodular minimization oracle by exploiting a dual formulation of the parametric line search problem and recent advances in cutting plane methods. We obtain a running time of $$O\bigl(n^2 \log(nM\|d\|_1)\cdot \text{EO} + n^3 \log(nM\|d\|_1)\bigr) + O(1)\cdot \text{SFM},$$ where $M = \|f\|_\infty$ and EO is the cost of evaluating $f$ at a set. Note that when $\log \|d\|_1 = O(\log (nM))$, this matches the current best weakly polynomial running time for submodular function minimization [Lee, Sidford, Wong '15], and therefore, one cannot hope to improve this running time. Our approach proceeds by deriving a dual formulation that minimizes the Lov\'asz extension $F$ over a hyperplane intersecting the unit hypercube, and then solving this dual problem approximately via cutting-plane methods, after which we round to the exact intersection using the integrality of $f$ and $d$.