Submodular set functions, matroids and the greedy algorithm: Tight worst-case bounds and some generalizations of the Rado-Edmonds theorem

Conforti, Michele; Cornuéjols, Gérard

doi:10.1016/0166-218x(84)90003-9

Cited by 306 publications

(307 citation statements)

References 7 publications

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“…In the literature, various problems related to (P), including the optimal allocation in combinatorial auctions to be explained later, have been discussed over decades [4,7,11,16,27,33,34]. Recently, Calinescu et al [3] proposed an elegant framework for the approximation of the problem (P), which is based on the pipage rounding technique developed by Ageev and Sviridenko [1].…”

Section: Resultsmentioning

confidence: 99%

On the Pipage Rounding Algorithm for Submodular Function Maximization — A View From Discrete Convex Analysis

Shioura

2009

Discrete Math. Algorithm. Appl.

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We consider the problem of maximizing a nondecreasing submodular set function under a matroid constraint. Recently, Calinescu et al. (2007) proposed an elegant framework for the approximation of this problem, which is based on the pipage rounding technique by Ageev and Sviridenko (2004), and showed that this framework indeed yields a (1 − 1/e)-approximation algorithm for the class of submodular functions which are represented as the sum of weighted rank functions of matroids. This paper sheds a new light on this result from the viewpoint of discrete convex analysis by extending it to the class of submodular functions which are the sum of M -concave functions. M -concave functions are a class of discrete concave functions introduced by Murota and Shioura (1999), and contain the class of the sum of weighted rank functions as a proper subclass. Our result provides a better understanding for why the pipage rounding algorithm works for the sum of weighted rank functions. Based on the new observation, we further extend the approximation algorithm to the maximization of a nondecreasing submodular function over an integral polymatroid. This extension has an application in multi-unit combinatorial auctions.

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Section: Resultsmentioning

confidence: 99%

On the Pipage Rounding Algorithm for Submodular Function Maximization — A View From Discrete Convex Analysis

Shioura

2009

Discrete Math. Algorithm. Appl.

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“…Conforti and Cornuéjols [34] showed that the greedy algorithm achieves at least 1 κt (1 − e −κt ) and 1 1+κt -approximations of the optimal solution for uniform and non-uniform matroids, respectively. Note that κ t ∈ [0, 1] for any submodular function, and the greedy algorithm is optimal when κ t = 0.…”

Section: A Related Workmentioning

confidence: 99%

Subspace Selection for Projection Maximization With Matroid Constraints

Zhang

Wang

Chong

et al. 2017

IEEE Trans. Signal Process.

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Abstract-Suppose that there is a ground set which consists of a large number of vectors in a Hilbert space. Consider the problem of selecting a subset of the ground set such that the projection of a vector of interest onto the subspace spanned by the vectors in the chosen subset reaches the maximum norm. This problem is generally NP-hard, and alternative approximation algorithms such as forward regression and orthogonal matching pursuit have been proposed as heuristic approaches. In this paper, we investigate bounds on the performance of these algorithms by introducing the notions of elemental curvatures. More specifically, we derive lower bounds, as functions of these elemental curvatures, for performance of the aforementioned algorithms with respect to that of the optimal solution under uniform and non-uniform matroid constraints, respectively. We show that if the elements in the ground set are mutually orthogonal, then these algorithms are optimal when the matroid is uniform and they achieve at least 1/2-approximations of the optimal solution when the matroid is non-uniform.

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“…curvature is 0) are a special case of submodular functions so it is reasonable to expect the greedy bound to be a continuous function of the curvature. In [13], bounds that include the curvature are presented for single matroid systems, uniform matroids and independence systems. For a system that is the intersection of p matroids the greedy bound is shown to be 1 p+κ .…”

Section: Approximationsmentioning

confidence: 99%

“…Approximation bounds exist for optimizing over a uniform matroid [9], any single matroid [10], an intersection of p matroids and, more generally, p-systems [11] as well as for the class of k-exchange systems [12]. Some bounds that include the dependence on curvature are evaluated in [13].…”

Section: Introductionmentioning

confidence: 99%

The maximum traveling salesman problem with submodular rewards

Jawaid

Smith

2013

2013 American Control Conference

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Abstract-In this paper, we look at the problem of finding the tour of maximum reward on an undirected graph where the reward is a submodular function, that has a curvature of κ, of the edges in the tour. This problem is known to be NP-hard. We analyze two simple algorithms for finding an approximate solution. Both algorithms require O(|V | 3 ) oracle calls to the submodular function. The approximation factors are shown to be (1 − κ) , respectively; so the second method has better bounds for low values of κ. We also look at how these algorithms perform for a directed graph and investigate a method to consider edge costs in addition to rewards. The problem has direct applications in monitoring an environment using autonomous mobile sensors where the sensing reward depends on the path taken. We provide simulation results to empirically evaluate the performance of the algorithms.

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Submodular set functions, matroids and the greedy algorithm: Tight worst-case bounds and some generalizations of the Rado-Edmonds theorem

Cited by 306 publications

References 7 publications

On the Pipage Rounding Algorithm for Submodular Function Maximization — A View From Discrete Convex Analysis

On the Pipage Rounding Algorithm for Submodular Function Maximization — A View From Discrete Convex Analysis

Subspace Selection for Projection Maximization With Matroid Constraints

The maximum traveling salesman problem with submodular rewards

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