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Submodular functions, matroids and certain polyhedra
, 2003
"... The viewpoint of the subject of matroids, and related areas of lattice theory, has always been, in one way or another, abstraction of algebraic dependence or, equivalently, abstraction of the incidence relations in geometric representations of algebra. Often one of the main derived facts is that all ..."
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Cited by 355 (0 self)
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The viewpoint of the subject of matroids, and related areas of lattice theory, has always been, in one way or another, abstraction of algebraic dependence or, equivalently, abstraction of the incidence relations in geometric representations of algebra. Often one of the main derived facts is that all bases have the same cardinality. (See Van der Waerden, Section 33.) From the viewpoint of mathematical programming, the equal cardinality of all bases has special meaning — namely, that every basis is an optimumcardinality basis. We are thus prompted to study this simple property in the context of linear programming. It turns out to be useful to regard “pure matroid theory”, which is only incidentally related to the aspects of algebra which it abstracts, as the study of certain classes of convex polyhedra. (1) A matroid M = (E,F) can be defined as a finite set E and a nonempty family F of socalled independent subsets of E such that (a) Every subset of an independent set is independent, and (b) For every A ⊆ E, every maximal independent subset of A, i.e., every basis of A, has the same cardinality, called the rank, r(A), of A (with respect to M). (This definition is not standard. It is prompted by the present interest). (2) Let RE denote the space of realvalued vectors x = [xj], j ∈ E. Let R+E = {x: 0 ≤ x ∈ RE}. (3) A polymatroid P in the space RE is a compact nonempty subset of R+E such that (a) 0 ≤ x0 ≤ x1 ∈ P = ⇒ x0 ∈ P. (b) For every a ∈ R+E, every maximal x ∈ P such that x ≤ a, i.e., every basis x of a, has the same sum j∈E xj, called the rank, r(a), of a (with respect to P).
Learning submodular functions
 IN PROCEEDINGS OF THE 43RD ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2011
"... Submodular functions are discrete functions that model laws of diminishing returns and enjoy numerous algorithmic applications that have been used in many areas, including combinatorial optimization, machine learning, and economics. In this work we use a learning theoretic angle for studying submodu ..."
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Cited by 27 (4 self)
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Submodular functions are discrete functions that model laws of diminishing returns and enjoy numerous algorithmic applications that have been used in many areas, including combinatorial optimization, machine learning, and economics. In this work we use a learning theoretic angle for studying
Maximizing nonmonotone submodular functions
 IN PROCEEDINGS OF 48TH ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS
, 2007
"... Submodular maximization generalizes many important problems including Max Cut in directed/undirected graphs and hypergraphs, certain constraint satisfaction problems and maximum facility location problems. Unlike the problem of minimizing submodular functions, the problem of maximizing submodular fu ..."
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Cited by 146 (18 self)
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Submodular maximization generalizes many important problems including Max Cut in directed/undirected graphs and hypergraphs, certain constraint satisfaction problems and maximum facility location problems. Unlike the problem of minimizing submodular functions, the problem of maximizing submodular
Approximating Submodular Functions Everywhere
"... Submodular functions are a key concept in combinatorial optimization. Algorithms that involve submodular functions usually assume that they are given by a (value) oracle. Many interesting problems involving submodular functions can be solved using only polynomially many queries to the oracle, e.g., ..."
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Cited by 45 (4 self)
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Submodular functions are a key concept in combinatorial optimization. Algorithms that involve submodular functions usually assume that they are given by a (value) oracle. Many interesting problems involving submodular functions can be solved using only polynomially many queries to the oracle, e
Applications Of Submodular Functions
, 1993
"... Submodular functions and related polyhedra play an increasing role in combinatorial optimization. The present surveytype paper is intended to provide a brief account of this theory along with several applications in graph theory and combinatorial optimization. ..."
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Cited by 26 (2 self)
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Submodular functions and related polyhedra play an increasing role in combinatorial optimization. The present surveytype paper is intended to provide a brief account of this theory along with several applications in graph theory and combinatorial optimization.
Lecture 10 (Submodular function)
, 2006
"... Submodular functions are the functions that frequently appear in connection with many combinatorial optimization problems. For instance, cut capacity functions of networks and rank functions of matroids are submodular functions. Learning about the properties of them helps our general and individua ..."
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Submodular functions are the functions that frequently appear in connection with many combinatorial optimization problems. For instance, cut capacity functions of networks and rank functions of matroids are submodular functions. Learning about the properties of them helps our general
Submodular Function Minimization
 BASED ON CHAPTER 7 OF THE HANDBOOK ON DISCRETE OPTIMIZATION [54] VERSION 3
, 2007
"... This survey describes the submodular function minimization problem (SFM); why it is important; techniques for solving it; algorithms by Cunningham [7, 11, 12], by Schrijver [69] as modified by Fleischer and Iwata [20], by Orlin [64], by Iwata, Fleischer, and Fujishige [45], and by Iwata [41, 43] for ..."
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Cited by 29 (0 self)
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This survey describes the submodular function minimization problem (SFM); why it is important; techniques for solving it; algorithms by Cunningham [7, 11, 12], by Schrijver [69] as modified by Fleischer and Iwata [20], by Orlin [64], by Iwata, Fleischer, and Fujishige [45], and by Iwata [41, 43
Submodular Functions are Noise Stable
"... We show that all nonnegative submodular functions have high noisestability. As a consequence, we obtain a polynomialtime learning algorithm for this class with respect to any product distribution on {−1, 1} n (for any constant accuracy parameter ɛ). Our algorithm also succeeds in the agnostic set ..."
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Cited by 21 (4 self)
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We show that all nonnegative submodular functions have high noisestability. As a consequence, we obtain a polynomialtime learning algorithm for this class with respect to any product distribution on {−1, 1} n (for any constant accuracy parameter ɛ). Our algorithm also succeeds in the agnostic
Maximization of Approximately Submodular Functions
"... Abstract We study the problem of maximizing a function that is approximately submodular under a cardinality constraint. Approximate submodularity implicitly appears in a wide range of applications as in many cases errors in evaluation of a submodular function break submodularity. Say that F is εap ..."
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Abstract We study the problem of maximizing a function that is approximately submodular under a cardinality constraint. Approximate submodularity implicitly appears in a wide range of applications as in many cases errors in evaluation of a submodular function break submodularity. Say that F is ε
On the Reducibility of Submodular Functions
"... Abstract The scalability of submodular optimization methods is critical for their usability in practice. In this paper, we study the reducibility of submodular functions, a property that enables us to reduce the solution space of submodular optimization problems without performance loss. We introdu ..."
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Abstract The scalability of submodular optimization methods is critical for their usability in practice. In this paper, we study the reducibility of submodular functions, a property that enables us to reduce the solution space of submodular optimization problems without performance loss. We
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