Optimal diagnosis procedures for k-out-of-n structures

Chang, Ming-Feng; Shi, Wei; Fuchs, W.K.

doi:10.1109/12.54850

Cited by 39 publications

(26 citation statements)

References 11 publications

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“…There is an elegant polynomial-time exact algorithm solving the Stochastic BFE problem for Boolean k-of-n functions (cf. Salloum (1979); Salloum and Breuer (1984); Ben-Dov (1981); Chang et al (1990)).…”

Section: A4 Proof Of Lemma 11 Approximation Bounds For Truncated Smentioning

confidence: 99%

Scenario Submodular Cover

Grammel¹,

Hellerstein²,

Kletenik

et al. 2017

Approximation and Online Algorithms

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Many problems in Machine Learning can be modeled as submodular optimization problems. Recent work has focused on stochastic or adaptive versions of these problems. We consider the Scenario Submodular Cover problem, which is a counterpart to the Stochastic Submodular Cover problem studied by Golovin and Krause (2011). In Scenario Submodular Cover, the goal is to produce a cover with minimum expected cost, where the expectation is with respect to an empirical joint distribution, given as input by a weighted sample of realizations. In contrast, in Stochastic Submodular Cover, the variables of the input distribution are assumed to be independent, and the distribution of each variable is given as input. Building on algorithms developed by Cicalese et al. (2014) and Golovin and Krause (2011) for related problems, we give two approximation algorithms for Scenario Submodular Cover over discrete distributions. The first achieves an approximation factor of O(log Qm), where m is the size of the sample and Q is the goal utility. The second, simpler algorithm achieves an approximation bound of O(log QW ), where Q is the goal utility and W is the sum of the integer weights. (Both bounds assume an integer-valued utility function.) Our results yield approximation bounds for other problems involving non-independent distributions that are explicitly specified by their support.

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Section: A4 Proof Of Lemma 11 Approximation Bounds For Truncated Smentioning

confidence: 99%

Scenario Submodular Cover

Grammel¹,

Hellerstein²,

Kletenik

et al. 2017

Approximation and Online Algorithms

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“…decision tree) that minimizes the expected cost of testing an individual item. There are polynomial-time algorithms for solving the MinCost problem for standard k-of-n testing [1,3,10,11].Kodialam was the first to study the MaxThroughput k-of-n testing problem, for the special case where k = 1 [8]. He gave a O(n 3 log n) algorithm for the problem.…”

mentioning

confidence: 99%

Max-Throughput for (Conservative) k-of-n Testing

Hellerstein¹,

Özkan²,

Sellie³

2015

Algorithmica

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We define a variant of k-of-n testing that we call conservative k-of-n testing. We present a polynomialtime, combinatorial algorithm for the problem of maximizing throughput of conservative k-of-n testing, in a parallel setting. This extends previous work of Kodialam and Condon et al., who presented combinatorial algorithms for parallel pipelined filter ordering, which is the special case where k = 1 (or k = n) [4,5,8]. We also consider the problem of maximizing throughput for standard k-of-n testing, and show how to obtain a polynomial-time algorithm based on the ellipsoid method using previous techniques. 1 In an alternative definition of k-of-n testing, the task is to determine whether at least k of the n tests have a value of 1. Symmetric results hold for this definition.MaxThroughput problems are closely related to MinCost problems [6,9]. In the MinCost problem for k-of-n testing, in addition to the probabilities p i , there is a cost c i associated with performing the i th test. The goal is to find a testing strategy (i.e. decision tree) that minimizes the expected cost of testing an individual item. There are polynomial-time algorithms for solving the MinCost problem for standard k-of-n testing [1,3,10,11].Kodialam was the first to study the MaxThroughput k-of-n testing problem, for the special case where k = 1 [8]. He gave a O(n 3 log n) algorithm for the problem. The algorithm is combinatorial, but its correctness proof relies on polymatroid theory. Later, Condon et al. studied the problem, calling it "parallel pipelined filter ordering". They gave two O(n 2 ) combinatorial algorithms, with direct correctness proofs [5].Our Results. In this paper, we extend the previous work by giving a polynomial-time combinatorial algorithm for the MaxThroughput problem for conservative k-of-n testing. Our algorithm can be implemented to run in time O(n 2 ), matching the running time of the algorithms of Condon et al. for 1-of-n testing. More specifically, the running time is O(n(log n + k) + o), where o varies depending on the output representation used; the algorithm can be modified to produce different output representations. We discuss output representations below.The MaxThroughput problem for standard k-of-n testing appears to be fundamentally different from its conservative variant. We leave as an open problem the task of developing a polynomial time combinatorial algorithm for this problem. We show that previous techniques can be used to obtain a polynomial-time algorithm based on the ellipsoid method. This approach could also be used to yield an algorithm, based on the ellipsoid method, for the conservative variant.Output Representation For the type of representation used by Condon et al. in achieving their O(n 2 ) bound, o = O(n 2 ). A more explicit representation has size o = O(n 3 ). We also describe a new, more compact output representation for which o = O(n).In giving running times, we follow Condon et al. and consider only the time taken by the algorithm to produce the output representation. We note, howe...

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“…A dynamic programming approach has been proposed in [15] for threshold functions. Bioch and Ibaraki [16] and Chang et al [17] propose polynomial time algorithms that produce optimal solutions for k-out-of-n system. These algorithms are generalized further in Boros and Unluyurt [18] by providing a generalized algorithm that is optimal for double regular systems (having identical components).…”

Section: Introductionmentioning

confidence: 99%

Multiobjective Optimization of a Port-of-Entry Inspection Policy

Young

Zhu

et al. 2010

IEEE Trans. Automat. Sci. Eng.

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At the port-of-entry containers are inspected through a specific sequence of sensor stations to detect the presence of nuclear materials, biological and chemical agents, and other illegal cargo. The inspection policy, which includes the sequence in which sensors are applied and the threshold levels used at the inspection stations, affects the probability of misclassifying a container as well as the cost and time spent in inspection. In this paper we consider a system operating with a Boolean decision function combining station results and present a multi-objective optimization approach to determine the optimal sensor arrangement and threshold levels while considering cost and time. The total cost includes cost incurred by misclassification errors and the total expected cost of inspection, while the time represents the total expected time a container spends in the inspection system. An example which applies the approach in a theoretical inspection system is presented.

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Optimal diagnosis procedures for k-out-of-n structures

Cited by 39 publications

References 11 publications

Scenario Submodular Cover

Scenario Submodular Cover

Max-Throughput for (Conservative) k-of-n Testing

Multiobjective Optimization of a Port-of-Entry Inspection Policy

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