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Submodular Optimization with Submodular Cover and Submodular Knapsack Constraints

by Rishabh Iyer, Jeff Bilmes , 2013
"... We investigate two new optimization problems — minimizing a submodular function subject to a submodular lower bound constraint (submodular cover) and maximizing a submodular function subject to a submodular upper bound constraint (submodular knapsack). We are motivated by a number of real-world appl ..."
Abstract - Cited by 14 (8 self) - Add to MetaCart
We investigate two new optimization problems — minimizing a submodular function subject to a submodular lower bound constraint (submodular cover) and maximizing a submodular function subject to a submodular upper bound constraint (submodular knapsack). We are motivated by a number of real

Fast algorithms for maximizing submodular functions

by Ashwinkumar Badanidiyuru - In SODA , 2014
"... There has been much progress recently on improved approximations for problems involving submodular ob-jective functions, and many interesting techniques have been developed. However, the resulting algorithms are often slow and impractical. In this paper we develop algorithms that match the best know ..."
Abstract - Cited by 13 (3 self) - Add to MetaCart
time. Our first result is a simple algorithm that gives a (1 − 1/e − )-approximation for a cardinality constraint using O(n log n ) queries, and a 1/(p + 2 ` + 1 + )-approximation for the intersection of a p-system and ` knapsack (linear) constraints using O ( n2 log 2 n ) queries. This is the first

THE SUBMODULAR KNAPSACK POLYTOPE

by Alper Atamtürk, Vishnu Narayanan - FORTHCOMING IN DISCRETE OPTIMIZATION , 2009
"... The submodular knapsack set is the discrete lower level set of a submodular function. The modular case reduces to the classical linear 0-1 knapsack set. One motivation for studying the submodular knapsack polytope is to address 0-1 programming problems with uncertain coefficients. Under various as ..."
Abstract - Cited by 1 (1 self) - Add to MetaCart
The submodular knapsack set is the discrete lower level set of a submodular function. The modular case reduces to the classical linear 0-1 knapsack set. One motivation for studying the submodular knapsack polytope is to address 0-1 programming problems with uncertain coefficients. Under various

Non-monotone submodular maximization under matroid and knapsack constraints

by Jon Lee, Vahab S. Mirrokni, Viswanath Nagarajan, Maxim Sviridenko - In Proc. 41th ACM Symp. on Theory of Computing , 2009
"... Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Un ..."
Abstract - Cited by 40 (1 self) - Add to MetaCart
-monotone submodular functions. In particular, for any constant k, we present a 1 k+2+ 1 k +ǫ-approximation for the submodular maximization problem under k matroid constraints, 1 k+ǫ and a ( 1 5 − ǫ)-approximation algorithm for this problem subject to k knapsack constraints (ǫ> 0 is 1 any constant). We improve

Maximizing non-monotone submodular functions

by Uriel Feige, Vahab S. Mirrokni, Jan Vondrák - IN PROCEEDINGS OF 48TH ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS , 2007
"... Submodular maximization generalizes many important problems including Max Cut in directed/undirected graphs and hypergraphs, certain constraint satisfaction problems and maximum facility location problems. Unlike the problem of minimizing submodular functions, the problem of maximizing submodular fu ..."
Abstract - Cited by 146 (18 self) - Add to MetaCart
functions is NP-hard. In this paper, we design the first constant-factor approximation algorithms for maximizing nonnegative submodular functions. In particular, we give a deterministic local search 1 2-approximation and a randomized-approximation algo-

Approximations for Monotone and Non-monotone Submodular Maximization with Knapsack Constraints

by Ariel Kulik, Hadas Shachnai, Tami Tamir
"... Submodular maximization generalizes many fundamental problems in discrete optimization, including Max-Cut in directed/undirected graphs, maximum coverage, maximum facility location and marketing over social networks. In this paper we consider the problem of maximizing any submodular function subject ..."
Abstract - Cited by 4 (0 self) - Add to MetaCart
subject to d knapsack constraints, where d is a fixed constant. We establish a strong relation between the discrete problem and its continuous relaxation, obtained through extension by expectation of the submodular function. Formally, we show that, for any non-negative submodular function, an α

Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels

by Quentin H. Spencer, A. Lee Swindlehurst, Martin Haardt - IEEE TRANS. SIGNAL PROCESSING , 2004
"... The use of space-division multiple access (SDMA) in the downlink of a multiuser multiple-input, multiple-output (MIMO) wireless communications network can provide a substantial gain in system throughput. The challenge in such multiuser systems is designing transmit vectors while considering the co-c ..."
Abstract - Cited by 371 (29 self) - Add to MetaCart
-channel interference of other users. Typical optimization problems of interest include the capacity problem—maximizing the sum information rate subject to a power constraint—or the power control problem—minimizing transmitted power such that a certain quality-of-service metric for each user is met. Neither

Near-optimal sensor placements in gaussian processes

by Andreas Krause, Ajit Singh, Carlos Guestrin, Chris Williams - In ICML , 2005
"... When monitoring spatial phenomena, which can often be modeled as Gaussian processes (GPs), choosing sensor locations is a fundamental task. There are several common strategies to address this task, for example, geometry or disk models, placing sensors at the points of highest entropy (variance) in t ..."
Abstract - Cited by 342 (34 self) - Add to MetaCart
information is NP-complete. To address this issue, we describe a polynomial-time approximation that is within (1 − 1/e) of the optimum by exploiting the submodularity of mutual information. We also show how submodularity can be used to obtain online bounds, and design branch and bound search procedures. We

Submodular Maximization with Cardinality Constraints

by Niv Buchbinder, et al. , 2014
"... We consider the problem of maximizing a (non-monotone) submodular function subject to a cardinality constraint. In addition to capturing well-known combinatorial optimization problems, e.g., Max-k-Coverage and Max-Bisection, this problem has applications in other more practical settings such as natu ..."
Abstract - Cited by 16 (2 self) - Add to MetaCart
We consider the problem of maximizing a (non-monotone) submodular function subject to a cardinality constraint. In addition to capturing well-known combinatorial optimization problems, e.g., Max-k-Coverage and Max-Bisection, this problem has applications in other more practical settings

A note on maximizing a submodular set function subject to a knapsack constraint

by Maxim Sviridenko
"... ..."
Abstract - Cited by 106 (2 self) - Add to MetaCart
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