Faster Parametric Submodular Function Minimization by Exploiting Duality

TL;DR

This paper introduces a new weakly polynomial-time algorithm for parametric submodular function minimization that reduces oracle calls by leveraging duality and cutting plane methods, matching the best known time bounds.

Contribution

It develops the first weakly polynomial algorithms for the problem by exploiting a dual formulation and recent cutting plane techniques, improving efficiency over previous methods.

Findings

Reduces submodular minimization oracle calls

Matches the best known weakly polynomial time bounds

Provides a dual formulation approach for the problem

Abstract

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 $d$.