Intersecting and crossing submodular functions 86 a tree representations of crossfree families 87 b crossing submodular functions 91 c intersecting submodular functions 101 3. Submodular functions have been the preferred form of discrete objectives for summarising text lin and bilmes 2011, images simon, snavely, and seitz 2007 and data subset selection wei, iyer. However, the interest for submodular functions is not limited to dis. Intersecting and crossingsubmodular functions 86 a tree representations of crossfree families 87 b crossingsubmodular functions 91 c intersectingsubmodular functions 101 3. Submodular function minimization grotschel, lovasz, schrijver 1981, 1988 iwata, fleischer, fujishige 2000 schrijver 2000 iwata 2003. 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. As a result, we deduce that on nonnegative functions, the random set algorithm has. A note on the budgeted maximization of submodular functions. There are specialized techniques for optimizing submodular functions. Submodularity has been an increasingly useful tool in machine learning in recent years.
Optimization of submodular functions tutorial lecture ii. Parallel double greedy submodular maximization xinghao pan 1 stefanie jegelka 1 joseph gonzalez 1 joseph bradley 1 michael i. 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. 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. A class of submodular functions for document summarization. The central processor could combine their solutions using.
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 capture the law of diminishing returns and can be viewed as a generalization of convexity to functions over the boolean cube. Nonnegative linear combinations of submodular functions are submodular, therefore the sum of all x is submodular homework. There is a strong case for electrical network topologists and submodular function theorists being aware of each others fields. We discuss some of these examples further in the following. Its a field dedicated to the optimization of submodular functions. Fpga based high performance doubleprecision matrix multiplication. Rigid network design via submodular set function optimization iman shames, tyler h. You may be thinking you dont have to be a genius to make that inference so ill expand a little. Submodular functions and electrical networks by maude. Bilmes applications of submodular functions in speech and nlp page 2 43.
Supermodularity and submodularity are also defined for functions defined over subsets of a larger set. Notes on graph cuts with submodular edge weights max planck. At rst glance it appears very similar to concavity, in other ways it resembles convexity. Overview of submodular minimization constrained submodular minimization constraint approximation hardness hardness ref vertex cover 2 ugc khot,regev 03 kunif. An analysis of approximations for maximizing submodular. Since submodu lar functions may be minimized exactly, and maximized approximately. The big picture submodular func ons electrical networks. Such functions arise in different areas, such as combinatorial optimization, machine learning and economics. Harveyy satoru iwataz vahab mirroknix abstract submodular functions are a key concept in combinatorial optimization. This algorithm, an extension of the nagamochiibaraki minimum cut algorithm as simplified by stoer and wagner m. Summers abstract we consider the problem of constructing networks that exhibit desirable algebraic rigidity properties, which can provide signi. The quality of any subset x umay be measured as the. Submodular functions and optimization volume 58 annals.
Applications of submodular functions in speech and nlp. Optimization of submodular functions tutorial lecture ii jan vondrak1 1ibm almaden research center san jose, ca jan vondrak ibm almaden submodular optimization tutorial 1 24. Minimizing submodular functions over families of sets. Purchase submodular functions and electrical networks, volume 54 1st edition. 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. Approximating submodular functions everywhere michel x. Think about functions applied to a set of sets of elements, you want to find t. The approximation algorithm for submodular maximization is an approximation for the cascade in independent activation model with same factor. Since submodular functions may be minimized exactly, and maximized approximately with some guarantees, in polynomial time, they readily lead to e. Wagner, a simple min cut algorithm, proceedings of the european.
It has widely been recognized that submodular functions play essential roles in efficiently solvable combinatorial optimization problems. Submodular functions and optimization, volume 58 2nd edition. 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. A note on concentration of submodular functions jan vondr ak may 16, 2010 abstract we survey a few concentration inequalities for submodular and fractionally subadditive functions of independent random variables, implied by the entropy method for selfbounding functions. Submodular functions and electrical networks, volume 54 1st. The power of these concentration bounds is that they are dimensionfree. Submodular optimization and approximation algorithms satoru iwata rims, kyoto university submodular functions cut capacity functions matroid rank functions entropy functions finite set. Scheduling to minimize power consumption using submodular. Fujishige, on the polyhedral approach to submodular functions. All such submodular functions f are called representatives. Given a monotone, submodular f, construct using polyn oracle queries a function f such that. Algorithms for approximate minimization of the di erence between submodular functions, with applications rishabh iyer dept. At rst glance it appears very similar to concavity, in other ways it.
For example, the problem of finding a maximum weight independent. Practical benefits and relation to deep neural networks. An analysis of approximations for maximizing submodular set. Symposium on modeling and optimization in mobile, ad hoc, and wireless networks wiopt, 2012 36. Narayanan, submodular functions and electrical networks, north. This might suggest maximizing a submodular function is easy. Submodular functions and electrical networks, volume 54.
We saw that the greedy algorithm, which picks elements in descending order of function value, gives good approximation guarantee. Submodular functions have been the preferred form of discrete objectives for summarising text lin and bilmes 2011, images simon, snavely, and seitz 2007 and. A few examples of such functions include cuts functions of graphs and hypergraphs, rank functions of matroids and covering functions. We are interested in the information retrieval problem of. Entropy functions information sources entropy of the joint distribution conditional mutual information. 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. Tom mccormick on submodular functions i next half hour. Submodular functions these functions can be minimized ef. In proceedings of the 22nd acm symposium on parallelism in algorithms and architectures spaa 10. Minimizing nonsubmodular functions with graph cuts a.
Algorithms for optimizing the ratio of submodular functions. Minimizing nonsubmodular functions with graph cutsa. Cardinality constrained submodular function maximization, which aims to select a subset of size at most k to maximize a monotone submodular utility function, is the key in many data mining and. Thomas mccormick satoru iwata sauder school of business, ubc cargese workshop on combinatorial optimization, septoct 20. Notes on greedy algorithms for submodular maximization. Submodular optimization and approximation algorithms.
Related polyhedra 102 a generalized polymatroids 102 b polypseudomatroids 106 c ternary semimodular polyhedra 112 3. Wagner, a simple min cut algorithm, proceedings of the european symposium on algorithms esa 94, lncs 855. Purchase submodular functions and optimization, volume 58 2nd edition. We show that they naturally represent the problem of document summarization, and show good results. Introduction the unprecedented growth in modern datasets coming from di erent sources and modalities such as images, videos, sensor data, social networks, etc. There is a strong case for electrical network topologists and submodular function theorists.
Submodular functions and electrical networks pdf free. Recently, submodular functions have also found immense utility in several real world problems in machine learning and. Learning with submodular functions archive ouverte hal. Submodular functions are a powerful class of functions on discrete sets. One approach is to attempt to learn the submodular function based on either queries of some form or based on data. An introduction to submodular functions and optimization. Thomas mccormick satoru iwata sauder school of business, ubc cargese workshop on combinatorial optimization.
If gis a linear function then gis also linear and hence f gis submodular if fis submodular and gis linear. Intuitively, a submodular function over the subsets demonstrates diminishing returns. Ramakrishnan received january 31, 1992 revised febrtary 8, 1995 we consider the problem of characterizing the minimum of a submodular function when the minimization is restricted to a family of subsets. Background and related work in the mapreduce model, there are m. The cost of a cut is measured by a submodular function f.
Learning submodular functions with applications to multi. Several hard combinatorial optimization problems can be posed in this framework. Submodular utility maximization for deadline constrained. Learnability, structure, and optimization mariaflorina balcanyand nicholas j. Finally, anyone looking for an additional level of abstraction from submodular functions should look at murotas monograph on discrete convex analysis mur03. They have been used in many areas, including combinatorial optimization, machine learning, and. Submodular functions are discrete functions that model laws of diminishing returns and enjoy numerous algorithmic applications. Scheduling to minimize power consumption using submodular functions. 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.
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