Submodular Maximization over Multiple Matroids via Generalized Exchange Properties

Lee, Jon; Sviridenko, Maxim; Vondrák, Jan

doi:10.1007/978-3-642-03685-9_19

Cited by 86 publications

(135 citation statements)

References 26 publications

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“…Further work by Lee, Sviridenko and Vondrák [23] improves the approximation ratio in the case of matroid constraints to 1/( +1+1/( −1)+ ). Feldman et al [14] attain this ratio for non-monotone maximization in -exchange systems.…”

Section: B Related Workmentioning

confidence: 95%

“…More recently, Lee, Sviridenko and Vondrák [23] consider the problem of both monotone and non-monotone submodular maximization subject to multiple matroid constraints, attaining a 1/( + )-approximation for monotone submodular maximization subject to ≥ 2 constraints using local search. Feldman et al [14] show that a local search algorithm attains the same bound for the related class of -exchange systems, which includes the intersection of strongly base orderable matroids, as well as the independent set problem in ( + 1)-claw free graphs.…”

Section: B Related Workmentioning

confidence: 99%

“…In the following sections, we shall repeatedly make use of the following general property of submodular functions, given in [23].…”

Section: The Algorithmmentioning

confidence: 99%

See 2 more Smart Citations

A Tight Combinatorial Algorithm for Submodular Maximization Subject to a Matroid Constraint

Filmus

Ward

2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

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We present an optimal, combinatorial 1-1/e approximation algorithm for monotone submodular optimization over a matroid constraint. Compared to the continuous greedy algorithm (Calinescu, Chekuri, Pal and Vondrak, 2008), our algorithm is extremely simple and requires no rounding. It consists of the greedy algorithm followed by local search. Both phases are run not on the actual objective function, but on a related non-oblivious potential function, which is also monotone submodular.In our previous work on maximum coverage (Filmus and Ward, 2011), the potential function gives more weight to elements covered multiple times. We generalize this approach from coverage functions to arbitrary monotone submodular functions. When the objective function is a coverage function, both definitions of the potential function coincide. The parameters used to define the potential function are closely related to Pade approximants of exp(x) evaluated at x = 1. We use this connection to determine the approximation ratio of the algorithm.

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Section: B Related Workmentioning

confidence: 95%

Section: B Related Workmentioning

confidence: 99%

See 1 more Smart Citation

A Tight Combinatorial Algorithm for Submodular Maximization Subject to a Matroid Constraint

Filmus

Ward

2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

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“…Proof: To show that max π V L (π) can be approximated in polynomial time, we prove that V L (π) is equivalent to a submodular function with two matroids constraints (Lemma 15) and for a given assignment π, the utility of V L (π) can be computed in polynomial time (Lemma 16). Given these, we can use the submodular optimization algorithm of Lee, Sviridenko, and Vondrák [14] under k = 2 matroid constraints to yield an approximation of k+1+ 1 k + = 7 2 + .…”

Section: A Constant-factor Approximation For the Linear Mlpmentioning

confidence: 99%

Markov Layout

Chierichetti

Kumar

Raghavan

2011

2011 IEEE 52nd Annual Symposium on Foundations of Computer Science

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Consider the problem of laying out a set of n images that match a query onto the nodes of a √ n× √ n grid. We are given a score for each image, as well as the distribution of patterns by which a user's eye scans the nodes of the grid and we wish to maximize the expected total score of images selected by the user. This is a special case of the Markov layout problem, in which we are given a Markov chain M together with a set of objects to be placed at the states of the Markov chain. Each object has a utility to the user if viewed, as well as a stopping probability with which the user ceases to look further at objects. This layout problem is prototypical in a number of applications in web search and advertising, particularly in an emerging genre of search results pages from major engines. In a different class of applications, the states of the Markov chain are web pages at a publishers website and the objects are advertisements.We study the approximability of the Markov layout problem. Our main result is an O(log n) approximation algorithm for the most general version of the problem. The core idea is to transform an optimization problem over partial permutations into an optimization problem over sets by losing a logarithmic factor in approximation; the latter problem is then shown to be submodular with two matroid constraints, which admits a constant-factor approximation. In contrast, we also show the problem is APX-hard via a reduction from CUBIC MAX-BISECTION.We then study harder variants of greater practical interest of the problem in which no gaps -states of M with no object placed on them -are allowed. By exploiting the geometry, we obtain an O(log 3/2 n) approximation algorithm when the digraph underlying M is a grid and an O(log n) approximation algorithm when it is a tree. These special cases are especially appropriate for our applications.

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“…Submodularity of functions over finite sets plays an important role in discrete optimization (see, e.g., [12], [13], [5], [18], [16], [2], [19], [20], [6], [7], [15], [21], [1], [9], and [10]). It has been shown that, under submodularity, the greedy strategy provides at least a constant-factor approximation to the optimal strategy.…”

Section: Introductionmentioning

confidence: 99%

Bounds for approximate dynamic programming based on string optimization and curvature

Liu

Chong

Pezeshki

et al. 2014

53rd IEEE Conference on Decision and Control

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In this paper, we will develop a systematic approach to deriving guaranteed bounds for approximate dynamic programming (ADP) schemes in optimal control problems. Our approach is inspired by our recent results on bounding the performance of greedy strategies in optimization of string functions over a finite horizon. The approach is to derive a string-optimization problem, for which the optimal strategy is the optimal control solution and the greedy strategy is the ADP solution. Using this approach, we show that any ADP solution achieves a performance that is at least a factor of β of the performance of the optimal control solution, characterized by Bellman's optimality principle. The factor β depends on the specific ADP scheme, as we will explicitly characterize. To illustrate the applicability of our bounding technique, we present examples of ADP schemes, including the popular rollout method.

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Submodular Maximization over Multiple Matroids via Generalized Exchange Properties

Cited by 86 publications

References 26 publications

A Tight Combinatorial Algorithm for Submodular Maximization Subject to a Matroid Constraint

A Tight Combinatorial Algorithm for Submodular Maximization Subject to a Matroid Constraint

Markov Layout

Bounds for approximate dynamic programming based on string optimization and curvature

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