Identifying the Most Connected Vertices in Hidden Bipartite Graphs Using Group Testing

Wang, Jianguo; Lo, Eric; Yiu, Man Lung

doi:10.1109/tkde.2012.178

Cited by 14 publications

(6 citation statements)

References 34 publications

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“…Theoretical computer science Group testing has been applied to classical problems in theoretical computer science, including pattern matching [45,109,137] and the estimation of high degree vertices in hidden bipartite graphs [196].…”

Section: Data Sciencementioning

confidence: 99%

Group Testing: An Information Theory Perspective

Aldridge¹,

Johnson²,

Scarlett³

2019

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The group testing problem concerns discovering a small number of defective items within a large population by performing tests on pools of items. A test is positive if the pool contains at least one defective, and negative if it contains no defectives. This is a sparse inference problem with a combinatorial flavour, with applications in medical testing, biology, telecommunications, information technology, data science, and more.In this monograph, we survey recent developments in the group testing problem from an information-theoretic perspective. We cover several related developments: efficient algorithms with practical storage and computation requirements, achievability bounds for optimal decoding methods, and algorithmindependent converse bounds. We assess the theoretical guarantees not only in terms of scaling laws, but also in terms of the constant factors, leading to the notion of the rate of group testing, indicating the amount of information learned per test. Considering both noiseless and noisy settings, we identify several regimes where existing algorithms are provably optimal or near-optimal, as well as regimes where there remains greater potential for improvement.In addition, we survey results concerning a number of variations on the standard group testing problem, including partial recovery criteria, adaptive algorithms with a limited number of stages, constrained test designs, and sublineartime algorithms.S 0 , S 1 partition of the defective set (Equation (4.4)) ı information density (Equation (4.6)) X 0,τ , X 1,τ sub-matrices of X corresponding to (S 0 , S 1 ) with |S 0 | = τ (Equation (4.14))τ ) (Equation (4.16)) 2. If the preceding test is negative, A contains no defective items, and we halt. If the test is positive, continue. 3. If A consists of a single item, then that item is defective, and we halt. Otherwise, pick half of the items in A, and call this set B. Perform a single test of the pool B. 4. If the test is positive, set A := B. If the test is negative, set A := A \ B. Return to Step 3. A BRIEF REVIEW OF ZERO-ERROR GROUP TESTINGObserve that S(i) is the set of tests containing item i, while S(K) is the set of positive tests when the defective set is K.Definition 1.10. A matrix X is called k-separable if the support unions S(L) are distinct over all subsets L ⊆ {1, 2, . . . , n} of size |L| = k.A matrix X is calledk-separable if the support unions S(L) are distinct over all subsets L ⊆ {1, 2, . . . , n} of size |L| ≤ k.Clearly, using a k-separable matrix as a test design ensures that group testing will provide different outcomes for each possible defective set of size k; thus,

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Section: Data Sciencementioning

confidence: 99%

Group Testing: An Information Theory Perspective

Aldridge¹,

Johnson²,

Scarlett³

2019

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“…• Star: This model allows queries that look at an arbitrary subset of the edges that are connected to a single vertex, i.e, queries of the form Q v,S = {(u, v) | u ∈ S}. This model was used in [26] and is equivalent to the demand model studied in [24].…”

Section: Query Modelsmentioning

confidence: 99%

“…Other papers have considered other tasks such as completely identifying the hidden graph (e.g., [18,19,5,4,6,11,1]), calculating or approximating the number of edges in it [9,12], or computing some function of the graph [26,8].…”

Section: Tasksmentioning

confidence: 99%

Finding a Hidden Edge

Kupfer¹,

Nisan²

2022

Preprint

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We consider the problem of finding an edge in a hidden undirected graph G = (V, E) with n vertices, in a model where we only allowed queries that ask whether or not a subset of vertices contains an edge. We study the non-adaptive model and show that while in the deterministic model the optimal algorithm requires n 2 queries (i.e., querying for any possible edge separately), in the randomized model Θ (n) queries are sufficient (and needed) in order to find an edge. In addition, we study the query complexity for specific families of graphs, including Stars, Cliques, and Matchings, for both the randomized and deterministic models. Lastly, for general graphs, we show a trade-off between the query complexity and the number of rounds, r, made by an adaptive algorithm. We present two algorithms with O rn 2/r and Õ rn 1/r sample complexity for the deterministic and randomized models, respectively.

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“…Interestingly, bipartite independent set queries are known to be stronger than independent set queries [BHN + 18, CLW20]. Other variants of bipartite independent set queries, where one of the sets is a singleton, have also been studied [BGMP19, BKKR13,WLY13]. While these algorithms are randomized and approximate, other work considers exact graph learning problems [AN19, AA05, ABK + 04].…”

Section: Related Work and Other Queriesmentioning

confidence: 99%

Vector-Matrix-Vector Queries for Solving Linear Algebra, Statistics, and Graph Problems

Rashtchian,

Woodruff,

Zhu

2020

Preprint

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We consider the general problem of learning about a matrix through vector-matrix-vector queries. These queries provide the value of u T M v over a fixed field F for a specified pair of vectors u, v ∈ F n . To motivate these queries, we observe that they generalize many previously studied models, such as independent set queries, cut queries, and standard graph queries. They also specialize the recently studied matrix-vector query model. Our work is exploratory and broad, and we provide new upper and lower bounds for a wide variety of problems, spanning linear algebra, statistics, and graphs. Many of our results are nearly tight, and we use diverse techniques from linear algebra, randomized algorithms, and communication complexity.

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Identifying the Most Connected Vertices in Hidden Bipartite Graphs Using Group Testing

Cited by 14 publications

References 34 publications

Group Testing: An Information Theory Perspective

Group Testing: An Information Theory Perspective

Finding a Hidden Edge

Vector-Matrix-Vector Queries for Solving Linear Algebra, Statistics, and Graph Problems

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