Azam Sheikh Muhammad

Competitive group testing and learning hidden vertex covers with minimum adaptivity

Suppose that we are given a set of n elements d of which have a property called defective. A group test can check for any subset, called a pool, whether it contains a defective. It is known that a nearly optimal number of O(d log (n/d)) pools in 2 stages (where tests within a stage are done in parallel) are sufficient, but then the searcher must know d in advance. Here we explore group testing strategies that use a nearly optimal number of pools and a few stages although d is not known beforehand. We prove a lower bound of O(log d log log d) stages and a more general pools vs. stages tradeoff. This is almost tight, since O(log d) stages are sufficient for a strategy with O(d log n) pools. As opposed to this negative result, we devise a randomized strategy using O(d log (n/d)) pools in 3 stages, with any desired success probability 1-epsilon. With some additional measures even 2 stages are enough. Open questions concern the optimal constant factors and practical implications. A related problem motivated by, e.g., biological network analysis is to learn hidden vertex covers of a small size k in unknown graphs by edge group tests. Does a given subset of vertices contain an edge?) We give a 1-stage strategy using O(k^3 log n) pools, with any parameterized algorithm for vertex cover enumeration as a decoder. During the course of this work we also provide a classification of types of randomized search strategies in general.

Randomized group testing both query-optimal and minimal adaptive

The classical group testing problem asks to determine at most d defective elements in a set of n elements, by queries to subsets that return Yes if the subset contains some defective, and No if the subset is free of defectives. By the entropy lower bound,

BOUNDS FOR NONADAPTIVE GROUP TESTS TO ESTIMATE THE AMOUNT OF DEFECTIVES

\log_2\sum_{i=0}^d{n\choose i}

Two New Perspectives on Multi-Stage Group Testing

tests, which is essentially d log2 n , are needed at least. We devise group testing strategies that combine two features: They achieve this optimal query bound asymptotically, with a factor 1+o (1) as n grows, and they work in a fixed number of stages of parallel queries. Our strategies are randomized and have a controlled failure probability, i.e., constant but arbitrarily small. We consider different settings (known or unknown d , probably correct or verified outcome), and we aim at the smallest possible number of stages. In particular, 2 stages are sufficient if d grows slowly enough with n , and 4 stages are sufficient if d =o (n ).

Bounds for nonadaptive group tests to estimate the amount of defectives

Group testing is the problem of finding d defectives in a set of n elements, by asking carefully chosen subsets (pools) whether they contain defectives. Strategies are preferred that use both a small number of tests close to the information-theoretic lower bound d log n, and a small constant number of stages, where tests in every stage are done in parallel, in order to save time. They should even work if d is not known in advance. In fact, one can succeed with O(d log n) queries in two stages, if certain tests are randomized and a constant failure probability is allowed. An essential ingredient of such strategies is to get an estimate of d within a constant factor. This problem is also interesting in its own right. It can be solved with O(log n) randomized group tests of a certain type. We prove that O(log n) tests are also necessary, if elements for the pools are chosen independently. The proof builds upon an analysis of the influence of tests on the searchers ability to distinguish between any two candidate numbers with a constant ratio. The next challenge is to get optimal constant factors in the O(log n) test number, depending on the prescribed error probability and the accuracy of d. We give practical methods to derive upper bound tradeoffs and conjecture that they are already close to optimal. One of them uses a linear programming formulation.

Strict group testing and the set basis problem

The group testing problem asks to find d≪n defective elements out of n elements, by testing subsets (pools) for the presence of defectives. In the strict model of group testing, the goal is to identify all defectives if at most d defectives exist, and otherwise to report that more than d defectives are present. If tests are time-consuming, they should be performed in a small constant number s of stages of parallel tests. It is known that a test number O(dlogn), which is optimal up to a constant factor, can be achieved already in s=2 stages. Here we study two aspects of group testing that have not found major attention before. (1) Asymptotic bounds on the test number do not yet lead to optimal strategies for specific n,d,s. Especially for small n we show that randomized strategies significantly save tests on average, compared to worst-case deterministic results. Moreover, the only type of randomness needed is a random permutation of the elements. We solve the problem of constructing optimal randomized strategies for strict group testing completely for the case when d=1 and s≤2. A byproduct of our analysis is that optimal deterministic strategies for strict group testing for d=1 need at most 2 stages. (2) Usually, an element may participate in several pools within a stage. However, when the elements are indivisible objects, every element can belong to at most one pool at the same time. For group testing with disjoint simultaneous pools we show that Θ(sd(n/d)1/s) tests are sufficient and necessary. While the strategy is simple, the challenge is to derive tight lower bounds for different s and different ranges of d versus n.

Summarizing Online User Reviews Using Bicliques

The classical and well-studied group testing problem is to find d defectives in a set of n elements by group tests, which tell us for any chosen subset whether it contains defectives or not. Strategies are preferred that use both a small number of tests close to the informationtheoretic lower bound d log n, and a small constant number of stages, where tests in every stage are done in parallel, in order to save time. They should even work if d is completely unknown in advance. An essential ingredient of such competitive and minimal-adaptive group testing strategies is an estimate of d within a constant factor. More precisely, d shall be underestimated only with some given error probability, and overestimated only by a constant factor, called the competitive ratio. The latter problem is also interesting in its own right. It can be solved with O(log n) randomized group tests of a certain type. In this paper we prove that Ω(log n) tests are really needed. The proof is based on an analysis of the influence of tests on the searchers ability to distinguish between any two candidate numbers with a constant ratio. Once we know this lower bound, the next challenge is to get optimal constant factors in the O(log n) test number, depending on the desired error probability and competitive ratio. We give a method to derive upper bounds and conjecture that our particular strategy is already optimal.

A Toolbox for Provably Optimal Multistage Strict Group Testing Strategies

Group testing is the problem to identify up to d defectives out of n elements, by testing subsets for the presence of defectives. Let t(n,d,s) be the optimal number of tests needed by an s-stage strategy in the strict group testing model where the searcher must also verify that at most d defectives are present. We start building a combinatorial theory of strict group testing. We compute many exact t(n,d,s) values, thereby extending known results for s=1 to multistage strategies. These are interesting since asymptotically nearly optimal group testing is possible already in s=2 stages. Besides other combinatorial tools we generalize d-disjunct matrices to any candidate hypergraphs, and we reveal connections to the set basis problem and communication complexity. As a proof of concept we apply our tools to determine almost all test numbers for n<10 and some further t(n,2,2) values. We also show t(n,2,2)<2.44*log n+o(log n).

New Constructions for Competitive and Minimal-Adaptive Group Testing

With vast amounts of text being available in electronic format, such as news and social media, automatic multi-document summarization can help extract the most important information. We present and evaluate a novel method for automatic extractive multi-document summarization. The method is purely combinatorial, based on bicliques in the bipartite word-sentence occurrence graph. It is particularly suited for collections of very short, independently written texts often single sentences with many repeated phrases, such as customer reviews of products. The method can run in subquadratic time in the number of documents, which is relevant for the application to large collections of documents.

Hypothesis-Driven Approaches to Multivariate Analysis of qPCR Data

Group testing is the problem of identifying up to d defectives in a set of n elements by testing subsets for the presence of defectives. Let t(n,d,s) be the optimal number of tests needed by an s-stage strategy in the strict group testing model where the searcher must also verify that no more than d defectives are present. We develop combinatorial tools that are powerful enough to compute many exact t(n,d,s) values. This extends the work of Huang and Hwang (2001) for s = 1 to multistage strategies. The latter are interesting since it is known that asymptotically nearly optimal group testing is possible already in s = 2 stages. Besides other tools we generalize d-disjunct matrices to any candidate hypergraphs, which enables us to express optimal test numbers for s = 2 as chromatic numbers of certain conflict graphs. As a proof of concept we determine almost all test numbers for n ≤ 10, and t(n,2,2) for some larger n.