In this paper, we give the first constantfactor approximation algorithm for maximizing any nonnegative submodular function subject to multiple matroid or knapsack constraints. Interlaced greedy algorithm for maximization of submodular. Submodular optimization problems and greedy strategies. An important part of designing greedy algorithms is proving that these greedy choices actually lead to a globally optimal solution. An online algorithm for maximizing submodular functions matthew streeter daniel golovin december 20, 2007. The following corollary will be very useful in analysing greedy algorithms involving submodular functions. Maximizing nonmonotone submodular set functions subject to. Informally, a greedy algorithm is an algorithm that makes locally optimal decisions, without regard for the global optimum. 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. Submodular functions and base polyhedra let e be a.
There is an algorithm that computes the minimum of any submodular function f. The greedy algorithm henceforth referred to as greedy is a natural heuristic for maximizing a monotone submodular function subject to certain constraints. A constantfactor approximation algorithm has also been obtained for maximizing monotone submodular function with a knapsack constraint see section 4. Notes on greedy algorithms for submodular maximization. Another example is the composition of any monotone modular function g. As stated earlier, our focus is on the performance of the greedy algorithm and its variants, described below, in solving some special cases of this general problem. A simple combinatorial algorithm for submodular function minimization satoru iwata. Second, we extend the double greedy technique, which achieves a tight 12 approximation for unconstrained submodular maximization, to the continuous setting. When z is integervalued and zo0, it is shown that the value of a greedy heuristic.
Maximizing a submodular set function subject to a matroid. Approximation algorithm for covering with arbitrary constraints and submodular cost. However, a simple greedy algorithm yields a constantfactor approximation guarantee by selecting a strategy with the maximum marginal contribution to the function value in each round 16, 17. 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.
While there are strong theoretical guarantees on the performance of greedy for maximizing submodular functions, there are few guarantees for non submodular ones. An online algorithm for maximizing submodular functions. Revisiting the greedy approach to submodular set function. As an example, the generalized assignment problem is a special case of the. We begin by analyzing the algorithm and then show how to implement it e ciently using a \threshold algorithm. It reuses the standard greedy algorithm of fisher et al. In particular, we show in section 3 that the forward regression and omp algorithms are within a. Request pdf a parallel double greedy algorithm for submodular maximization we study parallel algorithms for the problem of maximizing a nonnegative submodular function. The greedy algorithm henceforth referred to as greedy is a natural heuristic for maximizing.
Submodular maximization with cardinality constraints. For the problem of maximizing a submodular function subject to a matroid constraint special case of p 1, the greedy algorithm achieves a ratio of 12. A unified continuous greedy algorithm for submodular. Theoretically, unconstrained sfm can be performed in polynomial time 10, 11. Discovering important properties of drsubmodular function, we propose a fast double greedy algorithm which improves the running time. Thanks for contributing an answer to computer science stack exchange. In several settings it provides good approximation ratios, and until quite recently, the approximation ratios provided by greedy. Provable submodular minimization using wolfes algorithm. If a submodular function fis monotone1, greedy approaches for mcc have proven effective and nearly optimal, both in terms of query complexity and approximation factor. How good is the greedy strategy compared to the optimal solution. Guarantees for greedy maximization of nonsubmodular. The function ve is submodular and nondecreasing see proposition 3. We investigate the performance of the standard greedy algorithm for cardinality constrained maximization of non submodular nondecreasing set functions.
We investigate the performance of the standard greedy algorithm for cardinality constrained maximization of nonsubmodular nondecreasing set functions. The greedy strategy is an approximation algorithm to solve optimization problems arising in decision making with multiple actions. Orlin y abstract this paper presents a new simple algorithm for minimizing submodular functions. For integer valued submodular functions, the algorithm runs in on6eolognm time, where n is the cardinality of the ground set, m is.
We saw that the greedy algorithm, which picks elements in descending order of function value, gives good approximation guarantee. The first is set submodular optimization, which is to choose a set of actions to. Monotone submodular maximization over a matroid via non. Maximizing a monotone submodular function subject to a. The natural greedy algorithm fails for even the simple cardinality. An approximation algorithm for distributed resilient. For example, when maximizing a monotone submodular function. Drsubmodular functions consider the case that we can choose multiple copies for each element in the ground set. The greedy algorithm yields an optimal solution to maxztx.
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. Unlike the continuous greedy algorithm, our algorithm requires. Siam journal on discrete mathematics siam society for. We emphasize that our results are for nonmonotone submodular functions. We obtain the strongest known theoretical performance guarantees for greedy algorithms for subset selection. 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. Unlike submodular minimization, submodular maximization is nphard. Submodularfunctionmaximization greedyalgorithmformonotonecase.
For instance max cut is a special case even when the function is required only to be nonnegative. Our online algorithm can be applied in environments where abstract jobs arrive one at a time, and one can complete the jobs by investing time in a number of abstract activities, according to some schedule. A deterministic approximation algorithm is presented for the maximization of non monotone submodular functions over a ground set of size n subject to cardi. The first is set submodular optimization, which is to. For integer valued submodular functions, the algorithm runs in on6eolognm time, where n is the cardinality of the ground set, m is the. We assume that the fraction of jobs completed by a schedule is a monotone, submodular function of. Sg is highly practical since it is fast, delivers high. A simple combinatorial algorithm for submodular function. Submodular optimization and approximation algorithms. It also generalizes many classical deterministic online algo. A simple greedy algorithm nevertheless gives a 1 1eapproximation to the problem of maximizing a monotone submodular function subject to a cardinality constraint. This generalization has many applications in machine learning.
The stochastic greedy algorithm sg is a randomized version of the greedy algorithm for submodular maximization with a size constraint. Maximizing nonmonotone submodular set functions subject. Streaming algorithms for submodular function maximization chandra chekuri shalmoli guptay kent quanrudz dept. One common way of formally describing greedy algorithms is in terms op. An analysis of the greedy algorithm for the submodular set covering problem springerlink. Pdf guarantees for greedy maximization of nonsubmodular.
But avoid asking for help, clarification, or responding to other answers. An analysis of the greedy algorithm for the submodular set. For a ground set v of size n, this greedy algorithm needs onk function eval. The algorithm starts with the empty set, and then repeats the following step for. Streaming algorithms for submodular function maximization. Multidocument summarization via budgeted maximization of. Submodular welfare generalization of the previous example.
Dr submodular functions consider the case that we can choose multiple copies for each element in the ground set. Orlin y june 2008 abstract this paper presents a new simple algorithm for minimizing submodular functions. Nemhauser,wolsey,fisher 78 pick elements onebyone, maximizing the gain in fs, while. Another important class of non submodular set functions. Submodular set function maximization unlike the case of minimization, maximizing a submodular functions is nphard even in the unconstrained setting. The greedy algorithm for monotone submodular maximization. A fast double greedy algorithm for nonmonotone drsubmodular. This greedy algorithm starts with the empty set a 0 and in iteration i, adds an element maximizing the marginal gain eja i 1. Submodular functions have a natural diminishing returns property which.
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. However, greedy enjoys strong empirical performance. An analysis of approximations for maximizing submodular. However, greedy enjoys strong empirical performance for many important non. The minimumnormpoint algorithm applied to submodular. To resolve this problem, most of the algorithms such as nfindr and sga in 6 use dimension. Greedy submodular maximization consider a set function fde. Greedy algorithms for subset selection, sparse approximation and dictionary selection 2. Sk is a monotone nondecreasing submodular set function1, the greedy algorithm enjoys the multiplicative approximation guarantee of 1 1e nemhauser et al. 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. In this paper, we study parallelizations of this algorithm. Discrete mathematics, algorithms and applications 12. Several algorithms for submodular optimization described in this survey are. In this case the approximation ratios obtained are 1 e cc and 1 e cc, respectively, again matching the performance of the continuous greedy algorithm 35.
In mathematics, a submodular set function also known as a submodular function is a set function whose value, informally, has the property that the difference in the incremental value of the function that a single element makes when added to an input set decreases as the size of the input set increases. Base polyhedra and submodular function minimization in this section we show how the wolfe algorithm can be used to minimize submodular functions. We take into account the similar settings of attacks and. Provable submodular minimization using wolfes algorithm deeparnab chakrabarty prateek jain pravesh kothariy abstract owing to several applications in large scale learning and vision problems, fast submodular function minimization sfm has become a critical problem. In this survey, we mainly consider two classes of optimization problems where the objective function is submodular. Submodular function optimization submodularfunctionmaximization greedyalgorithmformonotonecase in. 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.876 1010 959 1330 308 1280 290 647 482 1575 911 1612 153 1528 1253 880 604 61 63 401 251 947 775 819 8 826 1383 223 326 416 259 318 1381 859 851 1563 962 618 457 1090 664 477 504 1389