There exists a fully submodular function, b f b f f i 0 such that provided that is nonempty. Nips 20 workshop on discrete optimization in machine learning. In particular, we examine the vertices and facets of these polyhedra. Submodular functions in graph theory georgia tech math. The adaptive complexity of maximizing a submodular function. In foundations of software technology and theoretical computer science.
Cuts in undirected graphs and hypergraphs yield symmetric submodular functions, satisfying fs. Submodular rate region models for multicast communication. Another important notion in graph theory is that of inde pendence. Submodularity1 is a property of set functions with deep theoretical. Convexity is a very natural property of various functions and domains occuring in such models. Submodular functions are relevant to machine learning for mainly two reasons. Submodular function maximization sfm has attracted much. Submodular functions often arise in various fields of operations research including discrete optimization, game theory, queueing theory and information theory. Submodular optimization and machine learning part 2. T1 an algorithm for submodular functions on graphs. Submodular functions in graph theory discrete mathematics.
Submodular function optimization file exchange matlab. Conjugate functions and a fencheltype minmax theorem for submodular and supermodular functions 199 a conjugate functions 199 b a fenchel. N2 a constructive method is described for proving the edmondsgiles theorem which. Submodular optimization and approximation algorithms satoru iwata rims, kyoto university submodular functions cut capacity functions matroid rank functions entropy functions finite. Empirically, this algorithm performs extremely well, but no theoretical analysis was given. Maximizing submodular functions part 2 windows on theory.
The present surveytype paper is intended to provide a brief account of this theory along with several applications in graph theory. In section 2 we give several equivalent definitions of submodular functions and. Continuing on my last post, today i will talk about recent work by niv buchbinder, moran feldman, seffi naor, and roy schwartz that gives a simple 12 approximation to the unconstrained. This toolbox provides functions for optimizing submodular set functions, i. In the standard notions of learnability for submodular functions such as pac val84 and its generalization pmac bh11, the input is a collection of sampled sets and their function values, and. We describe various aspects of the use of submodular functions in graph theory. Submodular functions in graph theory 237 to see the sufficiency, extend g by a minimum number of edges sv vev so as to have a required orientation. On the convergence rate of decomposable submodular. Theory and applications organized by stefanie jegelka, andreas krause, pradeep ravikumar, jeff bilmes. Convex functions occur in many mathematical models in economy, engineering, and other sciencies. Supermodularity and submodularity are also defined for functions. Finally, anyone looking for an additional level of abstraction from submodular. Submodularity functions exhibit su cient structure so that a mathematically beautiful and practically useful theory can be developed. Curvature and optimal algorithms for learning and minimizing submodular functions rishabh iyer y, stefanie jegelka, jeff bilmes yuniversity of washington, dept.
Submodular functions in graph theory sciencedirect. Annals of discrete mathematics submodular functions and. Among these setfunctions, submodular functions play an important role, similar to convex functions on vector spaces. Since the publication of the 1st edition of this book fifteen years ago, submodular functions have been showing further increasing importance in optimization, combinatorics, discrete mathematics. New proofs of theorems of mader and of tutte are provided as well as a new application on making a diagraph kedgeco. New proofs of theorems of mader and of tutte are provided as well as a new application on making a digraph. Survey on submodular function maximization by daniel golovin and andreas krause. Algorithm 1 greedy submodular function maximization with budget constraint. Learning with submodular functions francis bach mlss. Mathworks is the leading developer of mathematical computing software. Algorithms for approximate minimization of the di erence between submodular functions, with applications rishabh iyer dept. This book proposes representations of multicast rate regions in wireless networks based on the mathematical concept of submodular functions, e. Theory and applications i simons institute for the. In this paper, we show that the algorithm converges linearly, and we provide upper and lower bounds on the rate of.
In mathematics, a submodular set function also known as a submodular function is a set function whose value, informally, has the property that the difference in the incremental value of the function that a single element makes when added to an input set decreases as the size of the input set increases. Submodular functions cut capacity functions matroid rank functions entropy functions finite set. Submodular functions occur in a variety of areas including graph theory, information theory, combinatorial optimization, stochastic processes and game theory. Submodular functions and optimization volume 58 annals. Sample graphs from nci1 dataset 17 where each graph represents a chemical compound. Algorithms for optimizing the ratio of submodular functions graph gthat maps from any subset of the objects x uto the set of words x wcontained in the objects. A supermodular utility function is often related to complementary goods. In machine learning, they emerge in different forms as the potential functions of graphical models, as the utility functions in active learning and sensing, in models of diversity, in structured sparse estimation or network inference. Submodular functions and optimization, volume 58 2nd edition. In this tutorial, the theory of submodular functions is presented, in a self. Journal of machine learning research open source software paper.
Continuous methods for submodular maximization microsoft. In this lecture, the focus is on submodular function in combinatorial optimizations. Algorithms for approximate minimization of the di erence. Algorithms for optimizing the ratio of submodular functions. Intersecting submodular functions intersecting submodular theorem lovasz 1977 there exists a fully submodular function such that. Matlab toolbox for submodular function optimization maintained by andreas krause. The rst class of submodular functions which was studied thoroughly was the class of matroid rank functions. For realvalued functions, it seems appropriate to change the model by ignoring smallmagnitude errors in the predicted values. Convex analysis and optimization with submodular functions. Chapter 2 contains the essential mathematical background. New proofs of theorems of mader and of tutte are provided as well as a new application on making a diagraph kedge. Submodular functions examples for submodular functions 1 d gx. Let p denote the indegree function of this orientation.
Practical approaches to hard problems this draft is for personal use only. Submodular functions occur in a variety of areas including graph theory, information theory, combinatorial optimization, stochastic processes. Although our results apply to arbitrary submodular functions, they are much sharper for nondecreasing submodular functions. There are theoretically and practically reasonably e cient methods to. An analysis of approximations for maximizing submodular. Submodular optimization and approximation algorithms. In this monograph, the theory of submodular functions is presented in a selfcontained way, with all results proved from. The impact of information in greedy submodular maximization. Approximate minimization of general submodular functions via iterative graph cuts. A submodular framework for graph comparison stanford university. In mathematics, a submodular set function also known as a submodular function is a set function whose value, informally, has the property that the difference in the incremental value of the function.
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