Submodular function minimization

Abstract

Submodular function minimization (SFM) is a fundamental discrete optimization problem which generalizes many well known problems, has applications in various fields, and can be solved in polynomial time. Owing to applications in computer vision and machine learning, fast SFM algorithms are highly desirable. The current fastest algorithms [36] run in O(n 2 log nM · EO + n 3 log O(1) nM) time and O(n 3 log 2 n·EO+n 4 log O(1) n) time respectively, where M is the largest absolute value of the function (as-suming the range is integers) and EO is the time taken to evaluate the function on any set. Although the best known lower bound on the query complexity is only Ω(n) [23], the current shortest non-deterministic proof [10] certifying the optimum value of a function requires Ω(n 2) function evaluations. The main contribution of this paper are subquadratic SFM algorithms. For integer-valued submodular functions, we give an SFM algorithm which runs in O(nM 3 log n · EO) time giving the first nearly linear time algorithm in any known regime. For real-valued submodular functions with range in [−1, 1], we give an algorithm which iñ O(n 5/3 · EO/ε 2) time returns an ε-additive approximate solution. At the heart of it, our algorithms are projected stochastic subgradient descent methods on the Lovasz extension of submodular functions where we crucially exploit submodularity and data structures to obtain fast, i.e. sublinear time subgradient updates. The latter is crucial for beating the n 2 bound-we show that algorithms which access only subgradients of the Lovasz extension, and these include the empirically fast Fujishige-Wolfe heuristic [48, 15] and the theoretically best cutting plane methods [36] , must make Ω(n) subgradient calls (even for functions whose range is {−1, 0, 1}).

A submodular measure and approximate Gomory-Hu theorem for packing odd trails

“…For a given G, H, the laminar family L H , rooted forest F H , and function f H can be constructed in polynomial time using known algorithms for minimizing submodular functions [45,40,24]. In fact, the simple definition of perimeter allows us to use considerably more efficient cut-based methods [34] to implement Theorem 2.2 in time O(n 2 m).…”

Separation and Extension of Cover Inequalities for Conic Quadratic Knapsack Constraints with Generalized Upper Bounds

“…OPT is a constrained minimization of a submodular function. For surveys of submodular function minimization we refer the reader to Fujishige (2005), and Iwata (2008).…”

Divide-and-Conquer Algorithms for Partitioning Hypergraphs and Submodular Systems

“…A minimum (s, t)-partition of a submodular system (V, f ) can be computed by minimizing (asymmetric) submodular function f ′ on V \ {t} which is defined with an enough large constant M by f ′ (X) = f V \{t} (X) if s ∈ X and f ′ (X) = f V \{t} (X) + M otherwise. See [7] for recent algorithmic development of the submodular function minimization. Gueyranne [12] showed that computing minimum (s, t)-partitions of symmetric submodular systems is as hard as minimizing (asymmetric) submodular functions.…”