Submodularity is an important  property of set functions with deep theoretical results  and various  applications. Submodular systems appear in many applicable area, for example machine learning, economics, computer vision, social science, game theory and combinatorial optimization.  Nowadays submodular functions optimization has been attracted by many researchers.  Pendant pairs of a symmetric...

Let E be a finite set, R be the set of real numbers and f:2E-+R be a symmetric submodular function. The pair (E,f) is called a symmetric submodular system. We examine the structures of symmetric submodular systems and provide a decomposition theory of symmetric submodular systems. The theory is a generalization of the decomposition theory of 2-connected graphs developed by Tutte and can be appl...

In the present paper we extend the out-of-kilter method for the ordinary minimum-cost flow problem to the submodular flow problem considered by Edmonds and Giles. From the algorithmic point of view, there is no essential difference between a submodular flow problem described by a submodular function on a crossing family and one described by a submodular function on a distributive lattice, if we...

Let G = (V,E) be an undirected graph, LG ∈ R ×V be the associated Laplacian matrix, and b ∈ R V be a vector. Solving the Laplacian system LGx = b has numerous applications in theoretical computer science, machine learning, and network analysis. Recently, the notion of the Laplacian operator LF : R V → 2 V for a submodular transformation F : 2 → RE+ was introduced, which can handle undirected gr...

Many data summarization applications are captured by the general framework of submodular maximization. As a consequence, a wide range of efficient approximation algorithms have been developed. However, when such applications involve sensitive data about individuals, their privacy concerns are not automatically addressed. To remedy this problem, we propose a general and systematic study of diffe...

How can we extract representative features from a dataset containing sensitive personal information, while providing individual-level privacy guarantees? Many data summarization applications are captured by the general framework of submodular maximization. As a consequence, a wide range of efficient approximation algorithms for submodular maximization have been developed. However, when such app...

The submodular flow problem is extended by considering a nonseparable cost function, which is assumed to enjoy a variant of the exchange property of the base polyhedron of a submodular system. Two optimality criteria are established, one in terms of potentials associated with vertices and the other in terms of negative cycles in an auxiliary graph. These are natural extensions of the well-known...

In this paper we investigate k-submodular functions. This natural family of discrete functions includes submodular and bisubmodular functions as the special cases k = 1 and k = 2 respectively. In particular we generalize the known Min-Max-Theorem for submodular and bisubmodular functions. This theorem asserts that the minimum of the (bi)submodular function can be found by solving a maximization...

Submodular functions are relevant to machine learning for at least two reasons: (1) some problems may be expressed directly as the optimization of submodular functions and (2) the Lovász extension of submodular functions provides a useful set of regularization functions for supervised and unsupervised learning. In this monograph, we present the theory of submodular functions from a convex analy...

A submodular polyhedron is a polyhedron associated with a submodular function. This paper presents a strongly polynomial time algorithm for line search in submodular polyhedra with the aid of a fully combinatorial algorithm for submodular function minimization. The algorithm is based on the parametric search method proposed by Megiddo.