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Approximation algorithms for correlated knapsacks and nonmartingale bandits
 CoRR
"... Abstract — In the stochastic knapsack problem, we are given a knapsack of size B, and a set of items whose sizes and rewards are drawn from a known probability distribution. To know the actual size and reward we have to schedule the item—when it completes, we get to know these values. The goal is to ..."
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Cited by 20 (6 self)
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Abstract — In the stochastic knapsack problem, we are given a knapsack of size B, and a set of items whose sizes and rewards are drawn from a known probability distribution. To know the actual size and reward we have to schedule the item—when it completes, we get to know these values. The goal is to schedule the items (possibly making adaptive decisions based on the sizes seen so far) to maximize the expected total reward of items which successfully pack into the knapsack. We know constantfactor approximations when (i) the rewards and sizes are independent, and (ii) we cannot prematurely cancel items after we schedule them. What if either or both assumptions are relaxed? Related stochastic packing problems are the multiarmed bandit (and budgeted learning) problems; here one is given several arms which evolve in a specified stochastic fashion with each pull, and the goal is to (adaptively) decide which arms to pull, in order to maximize the expected reward obtained after B pulls in total. Much recent work on this problem focuses on the case when the evolution of each arm follows a martingale, i.e., when the expected reward from one pull of an arm is the same as the reward at the current state. What if the rewards do not form a martingale? In this paper, we give O(1)approximation algorithms for the stochastic knapsack problem with correlations and/or cancellations. Extending the ideas developed here, we give O(1)approximations for MAB problems without the martingale assumption. Indeed, we can show that previously proposed linear programming relaxations for these problems have large integrality gaps. So we propose new timeindexed LP relaxations; using a decomposition and “gapfilling ” approach, we convert these fractional solutions to distributions over strategies, and then use the LP values and the time ordering information from these strategies to devise randomized adaptive scheduling algorithms. 1.
Approximation algorithms for optimal decision trees and adaptive TSP problems
, 2010
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Raytheon BBN Technologies
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All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.
Approximation Algorithms for Correlated Knaspacks and NonMartingale Bandits
"... In the stochastic knapsack problem, we are given a knapsack of size B, and a set of jobs whose sizes and rewards are drawn from a known probability distribution. However, the only way to know the actual size and reward is to schedule the job—when it completes, we get to know these values. How should ..."
Abstract
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In the stochastic knapsack problem, we are given a knapsack of size B, and a set of jobs whose sizes and rewards are drawn from a known probability distribution. However, the only way to know the actual size and reward is to schedule the job—when it completes, we get to know these values. How should we schedule jobs to maximize the expected total reward? We know constantfactor approximations for this problem when we assume that rewards and sizes are independent random variables, and that we cannot prematurely cancel jobs after we schedule them. What can we say when either or both of these assumptions are changed? The stochastic knapsack problem is of interest in its own right, but techniques developed for it are applicable to other stochastic packing problems. Indeed, ideas for this problem have been useful for budgeted learning problems, where one is given several arms which evolve in a specified stochastic fashion with each pull, and the goal is to pull the arms a total of B times to maximize the reward obtained. Much recent work on this problem focus on the case when the evolution of the arms follows a martingale, i.e., when the expected reward from the future is the same as the reward at the current state. What can we say when the rewards do not form a martingale? In this paper, we give constantfactor approximation algorithms for the stochastic knapsack problem with
Raytheon BBN Technologies
"... Abstract—Physical phenomena such as temperature, humidity, and wind velocity often exhibit both spatial and temporal correlation. We consider the problem of tracking the extremum value of a spatiotemporally correlated field using a wireless sensor network. Determining the extremum at the fusion cen ..."
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Abstract—Physical phenomena such as temperature, humidity, and wind velocity often exhibit both spatial and temporal correlation. We consider the problem of tracking the extremum value of a spatiotemporally correlated field using a wireless sensor network. Determining the extremum at the fusion center after making all sensor nodes transmitting their measurements is not energyefficient because the spatiotemporal correlation of the field is not exploited. We present an optimal centralized algorithm that utilizes the aforementioned correlation to not only minimize the number of transmitting sensors but also ensure low tracking error with respect to the actual extremum. We use recent order statistics bounds in the formulation of the cost function. Since the centralized algorithm has high time complexity, we propose a suboptimal distributed algorithm based on a modified cost function. Our simulations indicate that a small fraction of sensors is often sufficient to track the extremum, and that the centralized algorithm can achieve about 70 % energy savings with almost perfect tracking. Furthermore, the performance of the distributed algorithm is comparable to that of the centralized algorithm with up to 25 % more energy expenditure. I.