Submodular functions and electrical networks by maude. This book has grown out of an attempt to understand the role that the topology of an electrical network plays in its efficient analysis. The subfolders contain source files for the experiments. For example, the problem of finding a maximum weight independent. We discuss some of these examples further in the following. Entropy functions information sources entropy of the joint distribution conditional mutual information.
To read the first four sections of this chapter the reader has t o be familiar with chapters 5, 6, 7 from the electrical networks part and the unstarred sections of the chapters on submodular functions. Scheduling to minimize power consumption using submodular functions. If gis a linear function then gis also linear and hence f gis submodular if fis submodular and gis linear. Narayanan, submodular functions and electrical networks, 1997. Revisiting the greedy approach to submodular set function.
Because generalized permutahedra can be constructed from submodular functions, and every generalized permutahedron has an associated submodular function, we have that there should be a correspondence between generalized permutahedra and polymatroids. Algorithms for approximate minimization of the di erence between submodular functions, with applications rishabh iyer dept. You may be thinking you dont have to be a genius to make that inference so ill expand a little. Practical benefits and relation to deep neural networks. In order to make the text useful to a wide audience, the material on electrical networks and that on submodular functions is presented independently of each other. Submodular utility maximization for deadline constrained. Its a field dedicated to the optimization of submodular functions. Submodular maximization generalizes many important problems including max cut in directed and undirected graphs and hypergraphs, certain constraint satisfaction problems and maximum facility location problems. Submodular function maximization via the multilinear.
Revisiting the greedy approach to submodular set function maximization pranava r. On submodularity and controllability in complex dynamical networks tyler h. Submodular function maximization has recently attracted considerable attention in theoretical computer science. Minimizing nonsubmodular functions with graph cuts a. Supermodularity and submodularity are also defined for functions defined over subsets of a larger set. We saw that the greedy algorithm, which picks elements in descending order of function value, gives good approximation guarantee. The big picture submodular func ons electrical networks. Since submodular functions may be minimized exactly, and maximized approximately with some guarantees, in polynomial time, they readily lead to e. Several hard combinatorial optimization problems can be posed in this framework. Tom mccormick on submodular functions i next half hour. For some submodular functions this problem can be solved exactly i.
In section 1, we give the different definitions of submodular functions. Electrical network theory download ebook pdf, epub. Distributed submodular maximization on massive datasets provable, constant factor, worstcase approximation guarantees. Algorithms for optimizing the ratio of submodular functions. All such submodular functions fare called representatives of f. While these results are wellknown in the optimization community, to our knowledge they were not used in the context of computer vision and mrf optimization. Think about functions applied to a set of sets of elements, you want to find t. Critically, our functions are monotone nondecreasing and submodular, which means that an efficient scalable greedy optimization scheme has a constant factor guarantee of optimality. In fact every polymatroid is a generalized permutahedron that has been translated to have a. Convexity is a very natural property of various functions and domains occuring in such models. There are specialized techniques for optimizing submodular functions.
Submodular functions have a natural diminishing returns property which makes them suitable for many applications, including approximation algorithms, game theory as functions modeling user preferences and electrical networks. Purchase submodular functions and optimization, volume 58 2nd edition. Submodular optimization and approximation algorithms. A class of submodular functions for document summarization. Nick harvey approximating submodular functions everywhere. An analysis of approximations for maximizing submodular. Chapter 12 dilworth truncation of submodular functions. Annals of discrete mathematics submodular functions and. Learning with submodular functions archive ouverte hal. A preliminary version of this paper entitled maximizing a submodular utility for deadline constrained data collection in sensor networks appeared in the proceedings of the 10th intl. Recently, submodular functions have also found immense utility in several real world problems in machine learning and. Presenting a topological approach to electrical network theory, this book demonstrates the strong links that exist between submodular functions and electrical networks. 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, algorithmic computer science, and algorithmic economics, and.
Our results on learning submodular functions are presented in. There is a strong case for electrical network topologists and submodular function theorists being aware of each others fields. Submodular functions are relevant to machine learning for at least two reasons. In our example, v may refer to the locations where sensors can be placed, and fs the utility. Narayanan, submodular functions and electrical networks, north. Minimizing nonsubmodular functions with graph cutsa. Convex analysis and optimization with submodular functions. As an example, for graph cut functions, it is easy to see that one can completely reconstruct the graph in on2 queries. An introduction to submodular functions and optimization. Convex functions occur in many mathematical models in economy, engineering, and other sciencies. Unlike the problem of minimizing submodular functions, the problem of maximizing submodular functions is nphard. All such submodular functions f are called representatives.
It has widely been recognized that submodular functions play essential roles in efficiently solvable combinatorial optimization problems. Maximizing a submodular set function subject to a matroid. Among these setfunctions, submodular functions play an important role, similar to convex functions on vector spaces. The emphasis in the book is on low complexity algorithms, particularly based on bipartite graphs. Submodular functions and electrical networks sciencedirect. In order to make the book useful to a wider audience, the material on electrical networks and that on submodular functions are presented independently of each other. This repository collects source code for the paper. Submodular optimization and approximation algorithms satoru iwata rims, kyoto university submodular functions cut capacity functions matroid rank functions entropy functions finite set.
Thomas mccormick satoru iwata sauder school of business, ubc cargese workshop on combinatorial optimization, septoct 20. A few examples of such functions include cuts functions of graphs and hypergraphs, rank functions of matroids and covering functions. Submodular functions and electrical networks, volume 54. Algorithms for approximate minimization of the di erence. In this tutorial, the theory of submodular functions is presented, in a selfcontained way, with all results shown from. A good knowledge of convex analysis is assumed see, e. Scheduling to minimize power consumption using submodular. Finding larger and larger inscribed ellipsoids informal statement i we have a. The quality of any subset x umay be measured as the. Presenting a topological approach to electrical network theory, this book demonstrates the strong links that exist.
For realvalued functions, it seems appropriate to change the model by ignoring smallmagnitude errors in the predicted values. Guarantees for greedy maximization of nonsubmodular functions. Background and related work in the mapreduce model, there are m. One approach is to attempt to learn the submodular function based on either queries of some form or based on data. Cortesi, and john lygeros abstractcontrollability and observability have long been recognized as fundamental structural properties of dynamical systems, but have recently seen renewed interest in the context of large, complex networks of. We are interested in the information retrieval problem of. 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. Submodular functions and optimization, volume 58 2nd edition. The hybrid rank problem, the bridge between topological electrical network theory and submodular functions, is covered in the final chapter. Submodular functions and optimization volume 58 annals. Hereby v is a nite set, commonly called the ground set.
Researchers interested in the link b e t w e e n submodular functions and electrical networks the key chapter for such a reader is chapter 14. Submodular function optimization file exchange matlab. Submodular functions and electrical networks pdf free. If, in addition, a u b f v, then a and b are crossing. A a n algorithm for submodular functions on graphs 99 family 9 of subsets of v is intersecting crossing if a n b, a u b e 9 for all intersecting crossing members a, b o 9. Submodular functions, optimization, and applications to.
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