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50
Maximizing nonmonotone submodular functions
 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 ..."
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Cited by 85 (13 self)
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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 functions is NPhard. In this paper, we design the first constantfactor approximation algorithms for maximizing nonnegative submodular functions. In particular, we give a deterministic local search 1 2approximation and a randomizedapproximation algo
Computer Experiments
, 1996
"... Introduction Deterministic computer simulations of physical phenomena are becoming widely used in science and engineering. Computers are used to describe the flow of air over an airplane wing, combustion of gasses in a flame, behavior of a metal structure under stress, safety of a nuclear reactor, a ..."
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Cited by 67 (5 self)
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Introduction Deterministic computer simulations of physical phenomena are becoming widely used in science and engineering. Computers are used to describe the flow of air over an airplane wing, combustion of gasses in a flame, behavior of a metal structure under stress, safety of a nuclear reactor, and so on. Some of the most widely used computer models, and the ones that lead us to work in this area, arise in the design of the semiconductors used in the computers themselves. A process simulator starts with a data structure representing an unprocessed piece of silicon and simulates the steps such as oxidation, etching and ion injection that produce a semiconductor device such as a transistor. A device simulator takes a description of such a device and simulates the flow of current through it under varying conditions to determine properties of the device such as its switching speed and the critical voltage at which it switches. A circuit simulator takes a list of devices and the
Nonmyopic active learning of gaussian processes: An explorationexploitation approach
 IN ICML
, 2007
"... When monitoring spatial phenomena, such as the ecological condition of a river, deciding where to make observations is a challenging task. In these settings, a fundamental question is when an active learning, or sequential design, strategy, where locations are selected based on previous measurements ..."
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Cited by 28 (3 self)
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When monitoring spatial phenomena, such as the ecological condition of a river, deciding where to make observations is a challenging task. In these settings, a fundamental question is when an active learning, or sequential design, strategy, where locations are selected based on previous measurements, will perform significantly better than sensing at an a priori specified set of locations. For Gaussian Processes (GPs), which often accurately model spatial phenomena, we present an analysis and efficient algorithms that address this question. Central to our analysis is a theoretical bound which quantifies the performance difference between active and a priori design strategies. We consider GPs with unknown kernel parameters and present a nonmyopic approach for trading off exploration, i.e., decreasing uncertainty about the model parameters, and exploitation, i.e., nearoptimally selecting observations when the parameters are (approximately) known. We discuss several exploration strategies, and present logarithmic sample complexity bounds for the exploration phase. We then extend our algorithm to handle nonstationary GPs exploiting local structure in the model. A variational approach allows us to perform efficient inference in this class of nonstationary models. We also present extensive empirical evaluation on several realworld problems.
Generalized MaximumEntropy Sampling
, 1999
"... We introduce the Generalized Constrained MaximumEntropy Sampling Problem (GCMESP) as a common generalization of the ordinary Constrained MaximumEntropy Sampling Problem (CMESP) and the Constrained DOptimality Problem (CDOPTP). Exact algorithms for both CMESP and CDOPTP are based on branchandbou ..."
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Cited by 26 (9 self)
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We introduce the Generalized Constrained MaximumEntropy Sampling Problem (GCMESP) as a common generalization of the ordinary Constrained MaximumEntropy Sampling Problem (CMESP) and the Constrained DOptimality Problem (CDOPTP). Exact algorithms for both CMESP and CDOPTP are based on branchandbound methods. We extend a spectral upperbounding method for the CMESP to the GCMESP. Introduction The Constrained DOptimality Problem (CDOPTP) and the Constrained MaximumEntropy Sampling Problem (CMESP) are both fundamental problems in experimental design. The CDOPTP has application in any statistical setting where we wish to fit a linear model and sampling is costly. The CMESP arises in such areas as environmental, geological and atmospheric monitoring, where establishing and maintaining monitoring devices is costly. In Section 1, we introduce the Generalized Constrained MaximumEntropy Sampling Problem (GCMESP) as a common generalization of the CMESP and the CDOPTP. In Section 2, generali...
Nonmyopic informative path planning in spatiotemporal models
, 2007
"... In many sensing applications we must continuously gather information to provide a good estimate of the state of the environment at every point in time. A robot may tour an environment, gathering information every hour. In a wireless sensor network, these tours correspond to packets being transmitted ..."
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Cited by 21 (8 self)
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In many sensing applications we must continuously gather information to provide a good estimate of the state of the environment at every point in time. A robot may tour an environment, gathering information every hour. In a wireless sensor network, these tours correspond to packets being transmitted. In these settings, we are often faced with resource restrictions, like energy constraints. The users issue queries with certain expectations on the answer quality. Thus, we must optimize the tours to ensure the satisfaction of the user constraints, while at the same time minimize the cost of the query plan. For a single timestep, this optimization problem is NPhard, but recent approximation algorithms with theoretical guarantees provide good solutions. In this paper, we present a new efficient algorithm, exploiting dynamic programming and submodularity of the information collected, that efficiently plans data collection tours for an entire (finite) horizon. Our algorithm can use any single step procedure as a black box, and, based on its properties, provides strong theoretical guarantees for the solution. We also provide an extensive empirical analysis demonstrating the benefits of nonmyopic planning in two real world sensing applications.
Sampling Strategies for Computer Experiments: Design and Analysis
, 2001
"... Computerbased simulation and analysis is used extensively in engineering for a variety of tasks. Despite the steady and continuing growth of computing power and speed, the computational cost of complex highfidelity engineering analyses and simulations limit their use in important areas like design ..."
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Cited by 20 (2 self)
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Computerbased simulation and analysis is used extensively in engineering for a variety of tasks. Despite the steady and continuing growth of computing power and speed, the computational cost of complex highfidelity engineering analyses and simulations limit their use in important areas like design optimization and reliability analysis. Statistical approximation techniques such as design of experiments and response surface methodology are becoming widely used in engineering to minimize the computational expense of running such computer analyses and circumvent many of these limitations. In this paper, we compare and contrast five experimental design types and four approximation model types in terms of their capability to generate accurate approximations for two engineering applications with typical engineering behaviors and a wide range of nonlinearity. The first example involves the analysis of a twomember frame that has three input variables and three responses of interest. The second example simulates the rollover potential of a semitractortrailer for different combinations of input variables and braking and steering levels. Detailed error analysis reveals that uniform designs provide good sampling for generating accurate approximations using different sample sizes while kriging models provide accurate approximations that are robust for use with a variety of experimental designs and sample sizes.
Nonmonotone submodular maximization under matroid and knapsack constraints
 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 ..."
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Cited by 17 (2 self)
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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. Unlike submodular minimization, submodular maximization is NPhard. In this paper, we give the first constantfactor approximation algorithm for maximizing any nonnegative submodular function subject to multiple matroid or knapsack constraints. We emphasize that our results are for nonmonotone submodular functions. In particular, for any constant k, we present a 1 k+2+ 1 k +ǫapproximation 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 the approximation guarantee of our algorithm to k+1+ 1 for k ≥ 2 k−1 +ǫ partition matroid constraints. This idea also gives aapproximation for maximizing a monotone submodular function subject to k ≥ 2 partition matroids, which improves over the previously best known guarantee of
Continuous Relaxations for Constrained MaximumEntropy Sampling
 In Integer Programming and Combinatorial Optimization
, 1996
"... . We consider a new nonlinear relaxation for the Constrained Maximum Entropy Sampling Problem  the problem of choosing the s \Theta s principal submatrix with maximal determinant from a given n \Theta n positive definite matrix, subject to linear constraints. We implement a branchandbound algo ..."
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Cited by 12 (8 self)
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. We consider a new nonlinear relaxation for the Constrained Maximum Entropy Sampling Problem  the problem of choosing the s \Theta s principal submatrix with maximal determinant from a given n \Theta n positive definite matrix, subject to linear constraints. We implement a branchandbound algorithm for the problem, using the new relaxation. The performance on test problems is far superior to a previous implementation using an eigenvaluebased relaxation. 1 Introduction Let n be a positive integer. For N := f1; : : : ; ng, let YN := fY j j j 2 Ng be a set of n random variables, with jointdensity function g N (\Delta). Let s be an integer satisfying 0 ! s n. For S ae N , j S j = s, let YS := fY j j j 2 Sg, and denote the marginal jointdensity function of YS by gS (\Delta). The entropy of S is defined by h(S) := \GammaE[ln gS (YS )]: Let m be a nonnegative integer, and let M := f1; 2; : : : mg. The Constrained MaximumEntropy Sampling Problem is then the problem of choosing a s...
Using Continuous Nonlinear Relaxations to Solve Constrained MaximumEntropy Sampling Problems
 Mathematical Programming, Series A
, 1998
"... We consider a new nonlinear relaxation for the Constrained MaximumEntropy Sampling Problem  the problem of choosing the s × s principal submatrix with maximal determinant from a given n × n positive definite matrix, subject to linear constraints. We implement a branchandbound algori ..."
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Cited by 11 (8 self)
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We consider a new nonlinear relaxation for the Constrained MaximumEntropy Sampling Problem  the problem of choosing the s × s principal submatrix with maximal determinant from a given n × n positive definite matrix, subject to linear constraints. We implement a branchandbound algorithm for the problem, using the new relaxation. The performance on test problems is far superior to a previous implementation using an eigenvaluebased relaxation. A parallel implementation of the algorithm exhibits approximately linear speedup for up to 8 processors, and has successfully solved problem instances that were heretofore intractable.
INTELLIGENT MAPS FOR AUTONOMOUS KILOMETERSCALE SCIENCE SURVEY
, 2008
"... We present a new approach for site survey by autonomous surface robots. In our method the agent constructs an intelligent map, a multiscale model of the explored environment incorporating in situ and remote sensing data. The agent learns the model’s parameters on the fly and exploits its prediction ..."
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Cited by 9 (4 self)
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We present a new approach for site survey by autonomous surface robots. In our method the agent constructs an intelligent map, a multiscale model of the explored environment incorporating in situ and remote sensing data. The agent learns the model’s parameters on the fly and exploits its predictions to guide adaptive navigation and sampling. In this manner the agent can respond appropriately to novel correlations, resource constraints and execution errors. Rover tests at Amboy Crater, California demonstrate improved performance over nonadaptive strategies for a geologic survey task.