At rst glance it appears very similar to concavity, in other ways it resembles convexity. What links here related changes upload file special pages permanent link. Since submodular functions may be minimized exactly, and maximized approximately with some guarantees, in polynomial time, they readily lead to e. It has widely been recognized that submodular functions play essential roles in efficiently solvable combinatorial optimization problems. Learning with submodular functions archive ouverte hal. Purchase submodular functions and optimization, volume 58 2nd edition. Distributed submodular maximization on massive datasets provable, constant factor, worstcase approximation guarantees. Minimizing nonsubmodular functions with graph cutsa. A few examples of such functions include cuts functions of graphs and hypergraphs, rank functions of matroids and covering functions. In particular, all the functions on set systems and. Submodular functions and electrical networks, volume 54 1st.
Fujishige, on the polyhedral approach to submodular functions. Submodular optimization and approximation algorithms satoru iwata rims, kyoto university submodular functions cut capacity functions matroid rank functions entropy functions finite set. We saw that the greedy algorithm, which picks elements in descending order of function value, gives good approximation guarantee. Presenting a topological approach to electrical network theory, this book demonstrates the strong links that exist between submodular functions and electrical networks. Pdf submodular functions are powerful tools to model and solve either to. Submodular function maximization has recently attracted considerable attention in theoretical computer science. Among these setfunctions, submodular functions play an important. 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. However, the interest for submodular functions is not limited to dis.
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. Intuitively, a submodular function over the subsets demonstrates diminishing returns. Several algorithms for submodular optimization described in this survey are implemented. In this talk we will give a brief overview of recent structural results about concise representations of submodular. Maximum likelihood bounded treewidth markov networks. For realvalued functions, it seems appropriate to change the model by ignoring smallmagnitude errors in the predicted values. Our results on learning submodular functions are presented in this new model, which we call the pmac model.
Submodular function optimization file exchange matlab central. Submodular functions satisfy a diminishing returns property that naturally applies to machine learning tasks such as data summarization, network monitoring, and clustering. Earlier work of bixby, cunningham, and topkis is extended to give a combinatorial algorithm for the problem of minimizing a submodular function, for which the amount of work is bounded by a polynomial in the size of the underlying set and the largest function value not its length. Presenting a topological approach to electrical network theory, this book demonstrates the strong links that exist. Nevertheless the deep submodular functions are not necessarily applicable. Rigid network design via submodular set function optimization iman shames, tyler h. 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. Examples of submodular functions are the rank functions of matroids and the cut functions, which are functions f given by a directed graph d v. A sum of weighted rank functions is also submodular.
Algorithms for approximate minimization of the di erence between submodular functions, with applications rishabh iyer dept. 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 book has grown out of an attempt to understand the role that the topology of an electrical network plays in its efficient analysis. Notes on graph cuts with submodular edge weights max planck. Algorithms for approximate minimization of the di erence. Our reduction illustrates the expressive power of submodular functions. Submodular functions and electrical networks by maude. Learning and testing submodular functions microsoft research. 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. Narayanan, submodular functions and electrical networks, north. Submodular functions capture the law of diminishing returns and can be viewed as a generalization of convexity to functions over the boolean cube.
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. Flow optimization in dynamic and continuous networks. Tom mccormick on submodular functions i next half hour. This might suggest maximizing a submodular function is easy. Submodular functions and optimization volume 58 annals. The tutorial is aimed at illustrating the importance of mathematics in vlsi, especially in the development. Finally, anyone looking for an additional level of abstraction from submodular functions should look at murotas monograph on. There are specialized techniques for optimizing submodular functions. Roughly speaking, the pmaclearning framework guarantees that the learned function is a constantfactor approximation of the true submodular function with high probability. Submodular functions and electrical networks, volume 54. Summers abstract we consider the problem of constructing networks that exhibit desirable algebraic rigidity properties, which can provide signi. Such functions arise in different areas, such as combinatorial optimization, machine learning and economics. In proceedings of the 22nd acm symposium on parallelism in algorithms and architectures spaa 10.
There is a strong case for electrical network topologists and submodular function theorists being aware of each others fields. All such submodular functions f are called representatives. Submodular functions and optimization, volume 58 2nd edition. First, we show that when marginal gains of submodular functions allow structured representations e. 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. Functions arising in this way form a rich class of submodular functions.
As examples, we construct submodular functions for three. Examples of submodular functions are the rank functions of matroids and the cut functions, which are functions f given by a. This algorithm, an extension of the nagamochiibaraki minimum cut algorithm as simplified by stoer and wagner m. In particular, we show that networks with desirable algebraic rigidity properties can be constructed using a simple greedy algorithm. This short introduction to submodular functions is meant for the second jcalm. Symposium on modeling and optimization in mobile, ad hoc, and wireless networks wiopt, 2012 36. Therefore, after learning a submodular function, one obtains an approximately submodular function.
A combinatorial algorithm minimizing submodular functions in. We consider the problem of maximizing submodular functions. Convex analysis and optimization with submodular functions. Minimizing symmetric submodular functions springerlink. A class of submodular functions for document summarization. You may be thinking you dont have to be a genius to make that inference so ill expand a little. Submodular optimization and approximation algorithms. Thomas mccormick satoru iwata sauder school of business, ubc cargese workshop on combinatorial optimization, septoct 20. Submodular functions and optimization volume 58 annals of. Scheduling to minimize power consumption using submodular. Its a field dedicated to the optimization of submodular functions. In 37, it provides a class of submodular functions used in the deep learning community called deep submodular functions.
Recently, submodular functions have also found immense utility in several real world problems in machine learning and. Background and related work in the mapreduce model, there are m. In order to apply the above ideas to dynamic networks with continuouslyvarying capacities, a continuous network is defined whose flows and capacities are additive set functions, and a generalization of the maxflow mincut theorem is proved. Submodular utility maximization for deadline constrained. Introduction the unprecedented growth in modern datasets coming from di erent sources and modalities such as images, videos, sensor data, social networks, etc. Algorithms for optimizing the ratio of submodular functions. An introduction to submodular functions and optimization. The hybrid rank problem, the bridge between topological electrical network theory and submodular functions, is covered in the final chapter. Moreover, we establish that certain cost functions that capture the algebraic properties of the networks are modular or submodular set functions, which. Submodular functions these functions can be minimized ef.
Wagner, a simple min cut algorithm, proceedings of the european symposium on algorithms esa 94, lncs 855. Submodular utility maximization for deadline constrained data. Supermodularity and submodularity are also defined for functions defined over subsets of a larger set. 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 utility functions in algorithmic game theory. Entropy functions information sources entropy of the joint distribution conditional mutual information. Maximizing submodular functions using probabilistic. The present surveytype paper is intended to provide a brief account of this theory along with. 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.
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. The quality of any subset x umay be measured as the. One function that has been featured prominently is the submodular facility location sfl function. Submodular function maximization via the multilinear. Finally, anyone looking for an additional level of abstraction from submodular functions should look at murotas monograph on discrete convex analysis mur03. Maximizing a submodular set function subject to a matroid. Submodular function maximization cmu school of computer. 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. In mathematics, a submodular set function is a set function whose value, informally, has the. 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.
Graph orientation there exists an arcconnected orientation. A combinatorial algorithm minimizing submodular functions. Unlike the problem of minimizing submodular functions, the problem of maximizing submodular functions is nphard. Minimizing nonsubmodular functions with graph cuts a. Submodular function optimization file exchange matlab. Crossing submodular functions crossing submodular theorem frank 1982, fujishige 1984 there exists a fully submodular function such that provided that is nonempty. We are interested in the information retrieval problem of. Submodular functions and related polyhedral play an increasing role in combinatorial optimization. Submodular maximization and diversity in structured output. At rst glance it appears very similar to concavity, in other ways it. Practical benefits and relation to deep neural networks.
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