Citeseerx submodular maximization by simulated annealing. Keywords and phrases submodular maximization, streaming algorithms, cardinality constraint. Submodular maximization by simulated annealing citeseerx. Deterministic algorithms for submodular maximization.
Tight approximation, minimum memory and low adaptive. The solution methodology for these formulations spanned across simulated annealing 2, branchandcut 15, and genetic algorithms 14. At first glance it appears very similar to concavity, in other ways it resembles convexity. Jul 04, 2011 in the nonmonotone submodular maximization problem we are given a nonnegative submodular function f, and the objective is to find a subset maximizing f. Maximizing nonmonotone submodular functions uriel feige. Submodular maximization with cardinality constraint. Pdf improving the neighborhood selection strategy in simulated. Vondrak, submodular maximization by simulated annealing, in proc. In the setting of constrained nonmonotone submodular maximization, lee et al. Submodular functions have a natural diminishing returns property which makes them suitable. Discrete algorithms society for industrial and applied mathematics, 2011. Finally, we discuss two naive approaches towards solving this problem. For this problem, there is an extremely simple algorithm that uses. Nonmonotone submodular maximization via a structural.
Building on the above two theorems, we provide two additional approximation. Mar 12, 2020 gharan so, vondrak j 2011 submodular maximization by simulated annealing. Submodular maximization has also recently attracted a signi cant amount of attention in the streaming and distributed settings lmsv11,kmvv15,mksk, bmkk14,mz15,benw15,benw16,cq19. Maxcut and one seeks constant factor approximations. Improved approximations for kexchange systems university of. Thus, a linear time algorithm is an algorithm which makes on oracle queries and on other operations, where n is the size of the ground set n. Randomization is widely used in submodular maximization. Nearly lineartime, parallelizable algorithms for non.
Additionally, one can nd submodular maximization problems in many other settings. The basic iteration of simulated annealing is as following. F lets i bethesetafteri iterationsofthegreedyalgorithm,thenby monotonicityfs fs. Submodular functions, optimization, and applications to. Discrete algorithms society for industrial and applied mathematics, 2011, pp.
Proof geman and geman have shown that a generic simulated annealing algorithm converges to a global optimum, if. In 4 it is further shown that continuous submodularity enables constantfactor approximation schemes for constrained monotone dr submodular maximization and box constrained nonmonotone submodular maximization problems. Extending the continuous greedy algorithm of 9 from the case of monotone submodular objectives to the general case of nonnegative submodular objectives which are not necessarily monotone, 20 obtained an improved approximation of 1e. Iyer et al, 20 fast semidi erential based submodular function optimization page 7 20. The maximization of this class of submodular function is much less explored than its monotone constrained counterpart. The problem has been extensively studied but most works focus on the submodular in. Submodular maximization and set systems play a major role in combinatorial. Since this class of problems includes maxcut, it is nphard.
Proceedings of the twentysecond annual acmsiam symposium on discrete algorithms. Submodular maximization by simulated annealing proceedings. Feldman, naor, and schwartz observed that if the simulated annealing algorithm of 15 outputs a relatively poor solution, then it must generate at some point a. We note that the distributed mapreduce model and adaptiv. Submodular maximization by simulated annealing stanford cs. In our formulation, the submodular function turns out to be nonnegative and nonmonotone. Some well known problems captured by unconstrained submodular maximization include maxcut, maxdicut, and variants of maxsat and maximum facility. Bilmes ee563spring 2018submodularity lecture 5 april 9th, 2018 f466 pg. Nonmonotone submodular function maximization under k. Oct 18, 2017 simulated annealing was first proposed in 1953 by metropolis et al. Bounds on doublesided myopic algorithms for unconstrained.
These results can be proved by considering the function. Maximizing submodular functions considerasetofelementsn f 1 ng andarealvaluedfunction overasubsetoftheelements f. Extending the continuous greedy algorithm of 8 from the case of monotone submodular objectives to the general case of nonnegative submodular objectives which are not necessarily monotone, 19 obtained an improved approximation of 1e o1. Vondrak, j submodular maximization by simulated annealing. We consider the problem of maximizing a nonnegative possibly nonmonotone submodular set function with or without constraints. For unconstrained maximization of general submodular functions, the problem is nphard to maximize e. Feldman, naor and schwarz later improved this to 0. Pdf submodular function maximization andreas krause. A framework for submodular optimization problems, called the contention resolution framework, was introduced recently by chekuri et al. Borodin 1 3, while the randomized version achieves 1 2 in expectation improving upon the 2 5 randomized localsearch approach in 10, and the 0. Extending the continuous greedy algorithm of 9 from the case of monotone submodular objectives to the general case of nonnegative submodular objectives which. A few recent works 7 4 have come up with a simulated annealing based randomized. In this paper, we propose a new algorithm for submodular maximization which is based on the idea of simulated annealing. Buchbinder, feldman, naor and schwarz have recently obtained an optimal 12 algorithm 5.
A 12approximation algorithm for maximizing a nonmonotone. Iyer et al, 20 fast semidi erential based submodular. A tight linear time 1 2 approximation for unconstrained. Acmsiam symposium on discrete algorithms soda, 2011. Positive influence maximization in signed social networks. Select a configuration choose a neighborhood compute the cost function if the cost is lowered, keep the configuration if it is higher, keep it only with a certain boltzmann probability the metropolis step reduce the temperature. Submodularfunctionmaximization greedyalgorithmformonotonecase. Theory and enumeration algorithms for finding local and global maxima minima of submodular supermodular functions can be found in b. The name and inspiration originate from annealing in metallurgy.
Nov 06, 2020 submodular maximization by simulated annealing proceedings of the twentysecond annual acmsiam symposium on discrete algorithms, society for industrial and applied mathematics, san francisco, california, usa 2011, pp. If fis submodular, then g is concave along all directions d 0. Continuous methods 21, 22 use a multilinear extension of the submodular set function to the convex hull. Submodular maximization by simulated annealing request pdf. Submodular maximization captures well known combinatorial optimization problems such as. Deterministic algorithms for submodular maximization problems. A concise description, motivation and implementation of each of these metaheuristics is given by brownlee 1. Fast semidifferential based submodular function optimization. If fis submodular, then g is convex in directions 1 i 1 j. In this paper, we propose a new algorithm for submodular maximization which is based on the idea of \em simulated annealing. The property of gthat it is concave in all directions d 0 makes it very appealing for ef.
A fast double greedy algorithm for nonmonotone drsubmodular. Thus, general purpose algorithms for the class tend to be approximation algorithms. Diverse particle selection is submodular maximization with cardinality constraint. As mentioned earlier, this problem has an obvious and natural social aspect but this aspect was paid no attention to until very recently when lappas et al 10 argued that apart from ensuring the required. A litany of questions from a wide variety of scientific disciplines can be cast as nonmonotone submodular maximization problems.
We prove that this algorithm achieves improved approximation for two problems. Jul 09, 2010 in this paper, we propose a new algorithm for submodular maximization which is based on the idea of \em simulated annealing. For unconstrained problem instances, one recent innovation in this vein includes an algorithm of buchbinder et al. An optimal streaming algorithm for nonmonotone submodular. A tight combinatorial algorithm for submodular maximization. Submodular maximization with nearly optimal approximation. A greedy algorithm is any algorithm that follows the problemsolving heuristic of making the locally optimal choice at each stage. In particular, many of the recent algorithms use an extension of submodular functions to fractional vectors known as the multilinear extension see, e.
Maxcut 32, 36, 39, 40, 53, maxdicut 20, 32, 34, generalized assignment 11, 12, 24, 29, and maxfacilitylocation 2, 14, 15. V as a set of actions which provide some bene t fs. Online continuous submodular maximization lin chen 1. We consider the problem of maximizing a nonnegative.
Submodular maximization with cardinality constraints. Some well known problems captured by unconstrained submodular maximization include maxcut, maxdicut, and variants of maxsat and maximum facility location. Guarantees of stochastic greedy algorithms for nonmonotone. Maximization of approximately submodular functions harvard seas. Submodular maximization by simulated annealing nasaads. Submodular maximization by simulated annealing core.
Pdf maximization of nonmonotone submodular functions. Submodular set function maximization unlike the case of minimization, maximizing a submodular functions is nphard even in the unconstrained setting. A tight linear time 12approximation for unconstrained. Nearoptimal map inference for determinantal point processes. Simulated annealing is the third most popular metaheuristic technique by number. Pdf submodular function maximization semantic scholar. Pdf submodularity in team formation madhavan pallan. Bilmes ee563spring 2018submodularity lecture 5 april 9th, 2018 f266 pg. Pdf pareto optimization for influence maximization in. 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.
Monotone submodular maximization over the bounded integer. Submodular maximization by simulated annealing shayan oveis gharan jan vondr aky abstract we consider the problem of maximizing a nonnegative possibly nonmonotone submodular set function with or without constraints. A fast algorithm for maximizing a nonmonotone drsubmodular. Thus, it is standard practice to assume that the function is accessed via a value oracle. Optimal streaming algorithms for submodular maximization with. In this paper, motivated by empirical evidences, we explore 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 globally optimal solution in a reasonable amount of time. This is one of the most basic submodular optimization problems, having a wide range of applications. Submodularity is a property of set functions with deep theoretical consequences and far reaching applications. In this realm, the evolving theory of submodular optimization has been a catalyst for progress in extraordinarily. Request pdf submodular maximization by simulated annealing we consider the problem of maximizing a nonnegative possibly nonmonotone submodular set function with or without constraints. Approximation for unconstrained submodular maximization. Simulated annealing is one the most well known local search methods.
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