The literature on constrained maximization of monotone submodular functions is very rich. Moreover, our algorithm easily extends to the maximization of a nonnegative continuous drsubmodular function subject to a box constraint, and achieves a tight 12approximation guarantee. Theneedsoftheapplications,andinparticularthesheer bulk of large data sets, have brought into focus the development of fast algorithms for. Submodular optimization and approximation algorithms satoru iwata rims, kyoto university. We do this by providing an example of a constraint family on a ground set of size n for which, if only given a linear optimization oracle, any algorithm for submodular maximization with a polynomial number of calls to the linear optimization oracle has an approximation ratio of only o log n v n log log n. The main result is for submodular functions that are \em nonmonotone. At rst glance it appears very similar to concavity, in other ways it resembles convexity.
Citeseerx document details isaac councill, lee giles, pradeep teregowda. Submodurlar function maximization approximation algorithms nemhauser, wolsey, fisher 1978. We consider maximizing a monotone submodular function under a cardinality constraint or a knapsack constraint in the streaming setting. Soma t, yoshida y 2017 nonmonotone drsubmodular function maximization. This paper presents a polynomialtime 12approximation algorithm for maximizing nonnegative ksubmodular functions. It is known that the greedy algorithm yields a 12 approximation 17 for this problem. Some of such utility functions are known to be approximately submodular. Siam journal on discrete mathematics siam society for. Adaptivity is not only a fundamental theoretical concept but it also has important practical consequences. Maximizing submodular or monotone approximately submodular. There is a wide variety of applications of submodular maximization where function evaluations are easily parallelized but each evaluation requires a long time to complete. Approximations for monotone and nonmonotone submodular maximization with knapsack constraints, mathematics of operations research, informs, vol. The minimization of ksubmodular functions is solvable in polynomial time 32.
Siam journal on discrete mathematics society for industrial. Discrete mathematics, algorithms and applications 12. Maximizing a monotone submodular function subject to a. As maximizing such a function is nphard, finding a maximum for such a function is often done using a greedy approach resulting in an approximate solution. There has been extensive work on constrained submodular function maximization when the functions are nonnegative. The quality of any subset x umay be measured as the. This approach resembles a common paradigm for designing approximation algorithms and is composed of two steps. Furthermore, there are online algorithms for submodular maximization 25, 26. On the other hand, some classes of submodular functions are deeply related to discrete concavity cf. Proceedings of the 31st aaai conference on artificial intelligence.
Several algorithms for submodular optimization described in this survey are. We study the problem of maximizing nonmonotone diminish return dr submodular function on the bounded integer lattice, which is a generalization of submodular set function. We present a recent improvement of the greedy algorithm with a much better asymptotic running time due to badanidiyuru and vondr ak 1. Typically, the approximation algorithms for these problems are based on either greedy algorithms or local search algorithms. Quanrudwork on this paper supported in part by nsf grants ccf19376, ccf1421231, and ccf1217462. A fast double greedy algorithm for nonmonotone drsubmodular. This makes it hard, for example, to apply standard derandomization techniques such as conditional expectations. Algorithms for optimizing the ratio of submodular functions. A deterministic approximation algorithm is presented for the maximization of non monotone submodular functions over a ground set of size n subject to cardi. We are given a set of elements over which a matroid is defined. Many of these algorithmic results exhibit a common structure. For the problem of maximizing an approximately submodular function asfm problem, a greedy algorithm quickly finds good feasible. Online algorithms for submodular minimization with. Constrained submodular maximization problems have long been studied, with nearoptimal results known under a variety of constraints when the submodular function is monotone.
A strongly polynomial algorithm has also been described in 10. There has been much progress recently on improved approximations for problems involving submodular objective functions, and many interesting techniques have been developed. Our focus is on nonmonotone submodular functions, i. Submodurlar function maximization approximation algorithms nemhauser, wolsey, fisher 1978 monotone sf cardinality constraint 11eapproximation vondrak stoc 2008 monotone sf matroid constraint 11eapproximation. At rst glance it appears very similar to concavity, in other ways it. 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. An online algorithm for maximizing submodular functions matthew streeter daniel golovin december 20, 2007. What is the adaptive complexity of maximizing a submodular function. An important open problem is the classical setting of matroid constraint, which has been instrumental for developments in. As in previous work, we measure the number of oracle calls to the objective function which is the dominating term in the running time. In this paper, we give the first constantfactor approximation algorithm for maximizing any nonnegative submodular function subject to multiple matroid or knapsack constraints.
Streaming algorithms for submodular function maximization chandra chekuri shalmoli guptay kent quanrudz dept. In this paper, we give constantfactor approximation algorithms for maximizing monotone k submodular. Therefore, a lot of work has been concerned with maximizing or minimizing a submodular function, often subject to combinatorial constraints. Interlaced greedy algorithm for maximization of submodular. This line of work was recently initiated by balkanski and singer and has already led to several interesting results on the cardinality constraint and explicit packing constraints. Cardinality constrained submodular function maximization, which aims to select a subset of size at most k to maximize a monotone submodular utility. Since 2012, more attention has been drawn to the case of general k. Fast algorithms for maximizing submodular functions.
Streaming algorithms for maximizing monotone submodular. In particular, the elements arrive sequentially and at any point of time, the algorithm has access to only a small fraction of the data stored in primary memory. Submodular maximization through the lens of linear. We are interested in the information retrieval problem of. Optimal algorithms for continuous nonmonotone submodular. Consider the following online version of the submodular maximization problem under a matroid constraint. Parallelizing greedy for submodular set function maximization. Fast algorithms for maximizing submodular functions proceedings. The goal is to incrementally choose a subset that remains independent in the matroid over time.
Nonmonotone drsubmodular function maximization aaai. Vondrak, maximizing a monotone submodular function subject to a matroid constraint, siam j. Studies on k submodular function optimization start with k 2, i. For the problem of maximizing an approximately submodular function asfm problem, a greedy.
Here, a na ve greedy algorithm nds a good approximation but is ine cient. Dr submodular functions consider the case that we can choose multiple copies for each element in the ground set. Our online algorithms could be used to select sensor placements when the same set of sensors is repeatedly deployed in an unknown or adversarial environment. Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including max cut in directedundirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. An important open problem is the classical setting of matroid constraint, which has been. For the problem without constraints as presented in definition 1, the goal is to maximize a nonmonotone submodular set function. Jun 20, 2015 we consider the problem of maximizing a nonnegative submodular set function \f. Moreover, our algorithm easily extends to the maximization of a nonnegative continuous drsubmodular function subject to a box constraint, and achieves a.
The problem of maximizing a nonnegative symmetric submodular function admits a 12 approximation algorithm. Previous work in this context has considered streaming algorithms for modular functions and monotone submodular functions. Derandomization of algorithms for general submodular function maximization seems hard since the access to the function is done via a value oracle. Buchbinderfeldmannaorschwartz2012 21 of unconstrained submodular minimization can be solved in polynomial time 32. For this problem, we know a wide variety of algorithms such as the wellknown greedy nemhauser et al. Namely, the function is extended to a continuous, usually nonlinear, function on a convex domain. The minimization of k submodular functions is solvable in polynomial time 32. The first works we are aware of to consider a onepass streaming algorithm for submodular maximization problems were the work of badanidiyuru et al. Deterministic algorithms for submodular maximization.
This family of optimization problems has several applications in machine learning, economics, and communication systems. Another line of work deals with maximizing normalized monotonesubmodular functions, again, subject to various. Streaming algorithms for submodular function maximization authors. Sampling algorithms, lower bounds submodular sparsest cut submodular load balancing. We consider parallel, or low adaptivity, algorithms for submodular function maximization. Computer science data structures and algorithms title. On the pipage rounding algorithm for submodular function. In other words, if are submodular on the same set, and are all nonnegative reals, then is also a submodular function on. Submodular functions play a fundamental role in combinatorial optimization, as they capture rank functions of. An online algorithm for maximizing submodular functions. In this paper we study the fundamental problems of maximizing a continuous nonmonotone submodular function over the hypercube, both with and without coordinatewise concavity.
The main result is for submodular functions that are non. Apr 26, 2019 when approaching to problems in computer science, we often encounter situations where a subset of a finite set maximizing some utility function needs to be selected. The case of nonmonotone submodular maximization is less understood. For submodular maximization, the intuition provided by the first definition is often. This generalization has many applications in machine learning. Fast algorithms for maximizing submodular functions dl. We consider the problem of maximizing a nonnegative submodular set function f. Unlike submodular minimization, submodular maximization is nphard. Citeseerx constrained nonmonotone submodular maximization.
Greedy algorithms are among the simplest and most intuitive algorithms known to humans. The problem of maximizing a given set function with a cardinality constraint has widespread applications. Streaming algorithms for submodular function maximization chandrachekuri shalmoligupta kent quanrud university of illinois at urbanachampaign october6,2015. Download citation a fast double greedy algorithm for nonmonotone dr submodular function maximization we study the problem of maximizing nonmonotone diminish return dr submodular function. In parallel to the progress in algorithms and complexity, submodularity has emerged as a useful property in machine learning. Definition 1 nonmonotone submodular function maximization without constraints. Some of the best situations in computer science are also when greedy algorithms are optimal or nearoptimal. In this paper, we give some theoretical results fundamental to the problem of finding a global maximum of a general submodular or, equivalently, global minimum of a general supermodular set function which we call the problem of maximization of submodular functions pmsf.
A simple combinatorial algorithm for submodular function. Feldman, deterministic algorithms for submodular maximization problems, acm trans. For the problem of maximizing an approximately submodular function asfm problem, a greedy algorithm quickly finds good feasible solutions for many. We consider the problem of maximizing a nonnegative submodular set function \f. At each time, a new weighted rank function of a different matroid. A collection of optimization algorithms for unconstrained submodular maximization usm of nonmonotone nonnegative set functions. Streaming algorithms for submodular function maximization.
In discrete optimization, submodular function is often regarded as discrete convexity, and indeed the maximization of a submodular function is known to be nphard. Recently, hazan and kale 12 developed algorithms for online and bandit submodular minimization without constraints c 2e. Deterministic algorithms for submodular maximization problems. By a set function we mean a mapping from 2 n to the real numbers, where n 1, 2. Notes on greedy algorithms for submodular maximization. A number of algorithms have been provided to solve the maximization problem when the set function is monotone and submodular. Best algorithms for approximating the maximum of a. Nonnegative submodular maximization in streams ok space feasibility oracle. Nov 30, 2018 we consider parallel, or low adaptivity, algorithms for submodular function maximization. Citeseerx streaming algorithms for submodular function.
Algorithmic methods for mining large graphs lecure 4. Approximation algorithms for online weighted rank function. Studies on ksubmodular function optimization start with k 2, i. Helsinki algorithms seminar, 10 jan 2014 deterministic algorithm. Monotone ksubmodular function maximization with size constraints.
In particular, nonnegative linear combinations of submodular functions are submodular. Submodular optimization and approximation algorithms. University of illinois at urbanachampaign october6,2015. When approaching to problems in computer science, we often encounter situations where a subset of a finite set maximizing some utility function needs to be selected. Pdf an online algorithm for maximizing submodular functions. Theory and enumeration algorithms article pdf available in european journal of operational research 1981. Recently, combinatorial strongly polynomial algorithms have been developed by 4, 12, 14, 19, 21.
Iwatafleischerfujishige schrijver, 2000 there is an algorithm that computes the minimum of any submodular function f. More recently, there is a large interest in constrained submodular function optimization driven both by theoretical progress and a variety of applicationsincomputerscience. However, realitybased set functions may not be submodular and may involve largescale and noisy data sets. Hub location as the minimization of a supermodular set function, operations research, informs, vol. 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. Jul 07, 2014 next, we can tweak and combine submodular functions to get more submodular functions. We emphasize that our results are for nonmonotone submodular functions. The adaptive complexity of maximizing a submodular function. Jan vondrak ibm almaden submodular functions and applications 9 28. Curvature and optimal algorithms for learning and minimizing. We also show that for monotone ksubmodular functions there is a polynomialtime k2k1approximation algorithm while for. We present an optimal, combinatorial 11e approximation algorithm for monotone submodular optimization over a matroid constraint.