Given a finite set V with n elements, a function f : 2 → Z is submodular if for all X,Y ⊆ V , f(X ∪ Y ) + f(X ∩ Y ) ≤ f(X) + f(Y ). Submodular functions frequently arise in combinatorial optimization. For example, the cut function in a weighted undirected graph and the rank function of a matroid are both submodular. Submodular function minimization is the problem of finding the global minimum o...

Many set functions F in combinatorial optimization satisfy the diminishing returns property F{A U X) — F(A) > F(A' U X) — F(A') for A C A!. Such functions are called submodular. A result from Nemhauser etal. states that the problem of selecting ^-element subsets maximizing a nondecreasing submodular function can be approximated with a constant factor (1 — 1/e) performance guarantee. Khuller eta...

Many combinatorial optimization problems have underlying goal functions that are submodular. The classical goal is to find a good solution for a given submodular function f under a given set of constraints. In this paper, we investigate the runtime of a multi-objective evolutionary algorithm called GSEMO until it has obtained a good approximation for submodular functions. For the case of monoto...

Submodular functions and related polyhedra play an increasing role in combinatorial optimization. The present survey-type paper is intended to provide a brief account of this theory along with several applications in graph theory and combinatorial optimization.

We study the problem of incorporating risk while making combinatorial decisions under uncertainty. formulate a discrete submodular maximization for selecting set using conditional value at (CVaR), metric commonly used in financial analysis. While CVaR has recently been optimization linear cost functions robotics, we take first step toward extending this to and provide several positive results. ...

Stochastic optimization of continuous objectives is at the heart of modern machine learning. However, many important problems are of discrete nature and often involve submodular objectives. We seek to unleash the power of stochastic continuous optimization, namely stochastic gradient descent and its variants, to such discrete problems. We first introduce the problem of stochastic submodular opt...

Submodular functions play a key role in combinatorial optimization and in the study of valued constraint satisfaction problems. Recently, there has been interest in the class of bisubmodular functions, which assign values to disjoint pairs of sets. Like submodular functions, bisubmodular functions can be minimized exactly in polynomial time and exhibit the property of diminishing returns common...

An M-convex function is a nonlinear discrete function defined on integer points introduced by Murota in 1996, and the M-convex submodular flow problem is one of the most general frameworks of efficiently solvable combinatorial optimization problems. It includes the minimum cost flow and the submodular flow problems as its special cases. In this paper, we first devise a successive shortest path ...