In many problems, a greedy strategy does not usually produce an optimal solution, but nonetheless a greedy heuristic may yield locally optimal solutions that approximate a. A unified continuous greedy algorithm for submodular. Another example is the composition of any monotone modular function g. While theoretically optimal, in practice these algorithms do not scale to large real world problems, since the inherently serial nature of the algorithms poses a challenge to leveraging advances in parallel hardware.
Probabilistic submodular maximization in sublinear time. Here, a na ve greedy algorithm nds a good approximation but is ine cient. Most classic results on the greedy algorithm and its variant assume the existence of. Data summarization in disguise marko mitrovic 1mark bun1 2 andreas krause3 amin karbasi abstract many data summarization applications are captured by the general framework of submodular maximization.
A deterministic approximation algorithm is presented for the maximization of nonmonotone submodular functions over a ground set of size nsubject to cardinality constraint k. Discovering important properties of drsubmodular function, we propose a fast double greedy algorithm which improves the running time. The greedy algorithm simply allocates each incoming item to the agent who gains from it the most and. Most classic results on the greedy algorithm and its variant assume the existence of an optimal polynomialtime incremental oracle that. Guarantees for greedy maximization of non submodular functions with applications andrew an biany 1joachim m. A fast double greedy algorithm for nonmonotone drsubmodular. Fischer nwf78 showed that a very natural greedy algorithm achieves a 1. Abstract submodular maximization is an important problem with many applications in engineering, computer science, economics and social sciences. An approximation algorithm for distributed resilient. This classic problem is known to be hard to approximate within factor better than 1. Request pdf a parallel double greedy algorithm for submodular maximization we study parallel algorithms for the problem of maximizing a nonnegative submodular function. The argument is reminiscent of a wellknown fact for submodular function maximization under cardinality constraints using the greedy. Guarantees for greedy maximization of nonsubmodular functions with applications andrew an biany 1joachim m.
The global objective is to maximize a submodular function on the strategy set with the existence of. One important result concerning submodularity in the in. In this paper, we present the stochasticlazier greedy algorithm slg to solve the corresponding non submodular maximization problem and offer a performance guarantee of the algorithm. A parallel double greedy algorithm for submodular maximization. An analysis of the greedy algorithm for the submodular set. Parallel double greedy submodular maximization github. Notes on greedy algorithms for submodular maximization. A parallel double greedy algorithm for submodular maximization alina ene. Strategic information sharing in greedy submodular. In this paper, we propose a 1 2approximation algorithm with a running time of o n log b, where n is the size of the ground set, b is the upper bound of integer lattice. Several algorithms for submodular optimization described in this survey are. For a general submodular function f with minimum value f 0. In the nonmonotone submodular maximization problem we are given a nonnegative submodular function f, and the objective is to. Submodular maximization with nearlyoptimal approximation.
Greedy algorithms are more common for maximizing monotone submodular functions. Greedy submodular maximization consider a set function fde. Naor roy schwartz technion israel institute of technology moran feldman, joseph sef. The authors also showed that current versions of the conditional gradient method a. Fast algorithms for maximizing submodular functions. An analysis of approximations for maximizing submodular.
The baseline algorithm in this setting is the greedy algorithm, due to fisher, nemhauser and wolsey, who initiated the study of problems involving maximization of submodular functions nwf78, fnw78, nw78. There has also been new improvements in the running time of the standard greedy solution for solving setcover a special case of submodular maximization when the data is. Constrained submodular maximization via greedy local. Like the continuous greedy algorithm, our algorithm delivers the optimal 1 1eapproximation. Greedy algorithm, answering this question armatively in the setting of the nonmonotone submodular maximization problem.
We improve on the best approximation factor known for this problem. The greedy algorithm simply allocates each incoming item to the agent who gains from it the. Typically, the approximation algorithms for these problems are based on either greedy algorithms or local search algorithms. Until recently, this algorithm had the best established performance guarantee for the problem with general matroid constraints. Adaptivity is not only a fundamental theoretical concept but it also has important practical consequences.
Submodularfunctionmaximization greedyalgorithmformonotonecase. Since the problem is nphard, a greedy algorithm has been developed, which gives an approximation within 12 of the optimal solution. Given the size of modern data sets, much work has focused on solving submodular maximization at scale. Naor technion roy schwartz technion abstractthe study of combinatorial problems with a submodular objective function has attracted much attention in recent years, and is partly motivated by the importance of such problems to economics. Suppose ss value is small compared to an optimal solution opt to the problem, yet s is structurally similar to opt. The algorithm starts with the empty set, and then repeats the following step for. Parallel double greedy submodular maximization xinghao pan 1 stefanie jegelka 1 joseph gonzalez 1 joseph bradley 1 michael i. Guarantees for greedy maximization of nonsubmodular.
A greedy algorithm is any algorithm that follows the problemsolving heuristic of making the locally optimal choice at each stage with the intent of finding a global optimum. In particular, we will cover the following 3 algorithms for maximizing f. As a concrete example, consider the maxcover problem. Kst09 recently showed that one could get essentially the same approximation subject to a constant number of knapsack constraints. It reuses the standard greedy algorithm of fisher et al. Our main result is an algorithm that achieves a nearlyoptimal 12.
Heuristics, greedy algorithm, interchange linear programming, matroid optimization, submodular set functions. Is it possible to maximize a monotone submodular function faster than the widely used lazy greedy algorithm also known as accelerated. This limitation raises the question of parallel algorithms for submodular maximization. Ccpv08 used the idea of multilinear extension of submodular functions and achieved optimal approximation algorithms for the problem of maximizing a monotone submodular function subject to a matroid.
Strategic information sharing in greedy submodular maximization. The greedy algorithm and other submodular maximization techniques are heavily used in machine learning and data mining since many fundamental objectives such as entropy, mutual information, graphs cuts, diversity, and set cover are all submodular. Revisiting the greedy approach to submodular set function. The adaptive complexity of maximizing a submodular function. The greedy algorithms that work well for centralized submodular optimization. Monotone submodular maximization over a matroid via non. Discovering important properties of dr submodular function, we propose a fast double greedy algorithm which improves the running time. Maximizing nonmonotone submodular set functions subject. Furthermore, if f is monotone submodular, s need not be a subset of t. Introduction in a recent paper cornuejols, fisher and nemhauser 2 give bounds on approximations heuristics and relaxations for the uncapacitated location prob lem.
The original continuous greedy algorithm 33,2 was not optimized for running time and required roughly on8 oracle calls to the objective function. Thus, no e cient approximation algorithm can be found for general submodular maximization. Recent work has shown that drsubmodular optimization problems have applications beyond submodular maximization 7, 6, 20, 5, including several of the applications mentioned above. A deterministic approximation algorithm is presented for the maximization of non monotone submodular functions over a ground set of size n subject to cardi. However, unlike the continuous greedy algorithm, our algorithm is entirely combinatorial, in the sense that it deals only with integral solutions to the problem. Consider a suboptimal solution s for a maximization problem. Therefore, we must use some smoothing methods for utilizing the established methods based on differentiation of outputs. Differentiable greedy submodular maximization with. This algorithm can be distributed among agents, each. This gives near optimal worst case approximation guarantees for the submodular maximization. The problem of maximizing a nonnegative symmetric submodular function admits a 12 approximation algorithm. Nonmonotone submodular maximization via a structural. Differentiable greedy submodular maximization with guarantees and gradient estimators 6 may 2020 shinsaku sakaue we consider making outputs of the greedy algorithm for monotone submodular function maximization differentiable w. What is the adaptive complexity of maximizing a submodular function.
The algorithm uses interlaced, thresholded greedy procedures to obtain tight ratio 14 in o n. The stochastic greedy algorithm sg is a randomized version of the greedy algorithm for submodular maximization with a size constraint. Probabilistic submodular maximization in sublinear time 2011. The optimal result for the wider, important class of nonmonotone functions an approximation guarantee of 12 is much more recent, and achieved by a double greedy algorithm by buchbinder et al. Distributed submodular maximization stanford computer science.
Conditional gradient method for stochastic submodular maximization. Sg is highly practical since it is fast, delivers high. Fast greedy algorithms in mapreduce and streaming ucsd cse. There has been extensive work on constrained submodular function maximization when the functions are nonnegative. Guarantees for greedy maximization of nonsubmodular functions with applications andrew an bian 1joachim m. Conditional gradient method for stochastic submodular. However, realitybased set functions may not be submodular and may involve largescale and noisy data sets. Recently, filmus and ward 11 developed a new algorithm, as well as a tighter. Buhmann andreas krause sebastian tschiatschek abstract we investigate the performance of the standard greedy algorithm for cardinality constrained maximization of nonsubmodular nondecreasing set functions.
In this paper we introduce the structural continuous greedy algorithm, answering this question. We saw that the greedy algorithm, which picks elements in descending order of function value, gives good approximation guarantee. In this paper, we present the stochasticlaziergreedy algorithm slg to solve the corresponding nonsubmodular maximization problem and offer a performance guarantee of the algorithm. A unified continuous greedy algorithm for submodular maximization. Submodular maximization with a dknapsack constraint arxiv.
Buhmann andreas krause sebastian tschiatschek abstract we investigate the performance of the standard greedy algorithm for cardinality constrained maximization of non submodular nondecreasing set functions. For example, in the kernelbased machine learning 1, 2, the. Interlaced greedy algorithm for maximization of 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. Guarantees for greedy maximization of non submodular functions with applications andrew an bian 1joachim m. We adopt a databasetransactional view of the serial double greedy algorithm which provides a provably optimal 12 approximation. This generalises earlier results of dobson and others on the applications of the greedy algorithm to the integer covering problem. An approximation algorithm for distributed resilient submodular maximization lifeng zhou and pratap tokekar abstractwe study a distributed resilient submodular maximization problem in which a group of robots collaboratively choose a strategy set. A na ve greedy algorithm is ine ective in this case, but we will.
As an example of an application in the datamining community, well look at diffusion. However, unlike the continuous greedy algorithm, our algorithm is entirely combinatorial, in the sense that it. Submodular maximization over multiple matroids via. Instead of local search, the double or bidirectional greedy algorithm maintains. The greedy algorithm is the best possible polynomialtime approach for solving a submodular function maximization with a nearoptimal approximation guarantee. The greedy algorithm is the best possible polynomialtime approach for solving a submodular function maximization with a nearoptimal. The proposed method avoids the requirement of a dimensionality reduction. We present a recent improvement of the greedy algorithm with a much better asymptotic running time due to badanidiyuru and vondr ak 1. Submodular maximization with nearlyoptimal approximation and. Develop submodular maximization algorithms that use fewer than evaluation queries avoid generic uses of the multilinear extension multilinear relaxation and rounding paradigm solve maxf x. The greedy algorithm for monotone submodular maximization. In many problems, a greedy strategy does not usually produce an optimal solution, but nonetheless a greedy heuristic may yield locally optimal solutions that approximate a globally optimal solution in a reasonable amount.
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