Submodular function optimization file exchange matlab. We are interested in the information retrieval problem of. A good knowledge of convex analysis is assumed see, e. As an example, for graph cut functions, it is easy to see that one can completely reconstruct the graph in on2 queries. If, in addition, a u b f v, then a and b are crossing. For more general submodular functions, we prove the following results. There is a strong case for electrical network topologists and submodular function theorists being aware of each others fields. Finally, anyone looking for an additional level of abstraction from submodular functions should look at murotas monograph on. Background and related work in the mapreduce model, there are m. When evaluated on duc 20042007 corpora, we obtain better than existing stateofart results in both generic and queryfocused document summarization. The emphasis in the book is on low complexity algorithms, particularly based on bipartite graphs.
Guarantees for greedy maximization of nonsubmodular functions. Submodular functions, optimization, and applications to. An introduction to submodular functions and optimization. Submodular optimization and approximation algorithms satoru iwata rims, kyoto university submodular functions cut capacity functions matroid rank functions entropy functions finite set. Algorithms for approximate minimization of the di erence. Think about functions applied to a set of sets of elements, you want to find t.
Maximizing a submodular set function subject to a matroid. Nick harvey approximating submodular functions everywhere. The quality of any subset x umay be measured as the. Submodular functions and electrical networks sciencedirect. In our example, v may refer to the locations where sensors can be placed, and fs the utility. Its a field dedicated to the optimization of submodular functions. Guarantees for greedy maximization of nonsubmodular functions with applications. Submodular optimization and approximation algorithms. Among these setfunctions, submodular functions play an important role, similar to convex functions on vector spaces. One approach is to attempt to learn the submodular function based on either queries of some form or based on data. Convexity is a very natural property of various functions and domains occuring in such models. For realvalued functions, it seems appropriate to change the model by ignoring smallmagnitude errors in the predicted values. Introduction the unprecedented growth in modern datasets coming from di erent sources and modalities such as images, videos, sensor data, social networks, etc.
A few examples of such functions include cuts functions of graphs and hypergraphs, rank functions of matroids and covering functions. All such submodular functions fare called representatives of f. The hybrid rank problem, the bridge between topological electrical network theory and submodular functions, is covered in the final chapter. Purchase submodular functions and optimization, volume 58 2nd edition.
Submodular functions and optimization volume 58 annals. 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. Supermodularity and submodularity are also defined for functions defined over subsets of a larger set. On submodularity and controllability in complex dynamical networks tyler h. The big picture submodular func ons electrical networks. 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. In fact every polymatroid is a generalized permutahedron that has been translated to have a. Scheduling to minimize power consumption using submodular functions.
Learning with submodular functions archive ouverte hal. Revisiting the greedy approach to submodular set function. 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. This book has grown out of an attempt to understand the role that the topology of an electrical network plays in its efficient analysis. In proceedings of the 22nd acm symposium on parallelism in algorithms and architectures spaa 10. 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. Submodular function maximization via the multilinear. If gis a linear function then gis also linear and hence f gis submodular if fis submodular and gis linear. 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. Tom mccormick on submodular functions i next half hour. In section 1, we give the different definitions of submodular functions. There are specialized techniques for optimizing submodular functions. 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. Submodular functions are relevant to machine learning for at least two reasons.
Several hard combinatorial optimization problems can be posed in this framework. We saw that the greedy algorithm, which picks elements in descending order of function value, gives good approximation guarantee. We discuss some of these examples further in the following. The subfolders contain source files for the experiments. 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. Revisiting the greedy approach to submodular set function maximization pranava r. Presenting a topological approach to electrical network theory, this book demonstrates the strong links that exist. You may be thinking you dont have to be a genius to make that inference so ill expand a little. Critically, our functions are monotone nondecreasing and submodular, which means that an efficient scalable greedy optimization scheme has a constant factor guarantee of optimality. Hereby v is a nite set, commonly called the ground set. 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. For some submodular functions this problem can be solved exactly i.
Convex functions occur in many mathematical models in economy, engineering, and other sciencies. Entropy functions information sources entropy of the joint distribution conditional mutual information. Fujishige, on the polyhedral approach to submodular functions. Submodular function maximization has recently attracted considerable attention in theoretical computer science. Unlike the problem of minimizing submodular functions, the problem of maximizing submodular functions is nphard. Scheduling to minimize power consumption using submodular. However, the interest for submodular functions is not limited to dis. Narayanan, submodular functions and electrical networks, north.
All such submodular functions f are called representatives. Intuitively, a submodular function over the subsets demonstrates diminishing returns. Finding larger and larger inscribed ellipsoids informal statement i we have a. 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. Thomas mccormick satoru iwata sauder school of business, ubc cargese workshop on combinatorial optimization, septoct 20. Recently, submodular functions have also found immense utility in several real world problems in machine learning and. Minimizing nonsubmodular functions with graph cutsa. At rst glance it appears very similar to concavity, in other ways it resembles convexity. Since submodular functions may be minimized exactly, and maximized approximately with some guarantees, in polynomial time, they readily lead to e. 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. It has widely been recognized that submodular functions play essential roles in efficiently solvable combinatorial optimization problems. Narayanan, submodular functions and electrical networks, 1997. Submodular functions and electrical networks by maude.
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. Submodular functions and optimization, volume 58 2nd edition. In this tutorial, the theory of submodular functions is presented, in a selfcontained way, with all results shown from. This is for a variety of reasons including applications, recognition of interesting algorithmic and structural properties, as well as the use of submodular functions as. A class of submodular functions for document summarization. Algorithms for optimizing the ratio of submodular functions. An analysis of approximations for maximizing submodular. Chapter 12 dilworth truncation of submodular functions.
Submodular utility maximization for deadline constrained. Distributed submodular maximization on massive datasets provable, constant factor, worstcase approximation guarantees. Algorithms for approximate minimization of the di erence between submodular functions, with applications rishabh iyer dept. Submodular functions and electrical networks pdf free. For example, the problem of finding a maximum weight independent. Practical benefits and relation to deep neural networks. Symposium on modeling and optimization in mobile, ad hoc, and wireless networks wiopt, 2012 36. This repository collects source code for the paper.
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