Abstract
In combinatorial group testing, there are n items; each has an unknown binary status, positive (i.e., defective) or negative (i.e., good), and the number of positives is upper bounded by an integer d. Suppose there is some method to test whether a subset of items contains at least one positive or not. The test result is said to be positive if it indicates that the subset contains at least one positive item; otherwise, the test result is called negative. The problem is to resolve the status of every item using the minimum number of tests.Group testing (GT) algorithms can be adaptive or nonadaptive. An adaptive algorithm conducts the tests one by one and allows to design later tests using the outcome information of all previous tests. A nonadaptive group testing (NGT) algorithm specifies all tests before knowing any test results, and the benefit is that all tests can be performed in parallel. For the above group testing problem, nonadaptive algorithms require inherently more tests than adaptive ones.Though the research of group testing dates back to Dorfman’s 1943 paper, a renewed interest in the subject occurred recently mainly due to the applications of group testing to the area of computational molecular biology. In applications of molecular biology, a group testing algorithm is called a pooling design, and the composition of each test is called a pool. While it is still important to minimize the number of tests, there are two other goals. First, in the biological setting, screening one pool at a time is far more expensive than screening many pools in parallel; this strongly encourages the use of nonadaptive algorithms. Second, DNA screening is error prone, so it is desirable to design error-tolerant algorithms, which can detect or correct some errors in the test results.In this monograph, some recent algorithmic, complexity, and mathematical results on nonadaptive group testing (and on pooling design) are presented.
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N. Alon, J.H. Spencer, The Probabilistic Method (Wiley, New York, 1992). (Second Edition 2000)
N. Alon, D. Moshkovitz, S. Safra, Algorithmic construction of sets for k-restrictions. ACM Trans. Algorithms 2(2), 153–177 (2006)
D.J. Balding, W.J. Bruno, E. Knill, D.C. Torney, A comparative survey of non-adaptive pooling designs, in Genetic Mapping and DNA Sequencing (Springer, New York, 1996), pp. 133–154
T. Berger, J.W. Mandell, P. Subrahmanya, Maximally efficient two-stage group testing. Biometrics 56, 833–840 (2000)
J. Borneman, M. Chrobak, G. Della Vedova, A. Figueroa, T. Jiang, Probe selection algorithms with applications in the analysis of microbial communities. Bioinformatics 17(Suppl.), S39–S48 (2001)
W.J. Bruno, D.J. Balding, E. Knill, D.C. Bruce et al., Efficient pooling designs for library screening. Genomics, 26, 21–30 (1995)
H.B. Chen, F.K. Hwang, Exploring the missing link among d-separable, \(\bar{d}\)-separable and d-disjunct matrices. Discret. Appl. Math. 133, 662–664 (2007)
J. Chen, I.A. Kanj, W. Jia, Vertex cover: further observations and further improvements. J. Algorithm 41(2), 280–301 (2001)
H.B. Chen, Y. Cheng, Q. He, C. Zhong, Transforming an error-tolerant separable matrix to an error-tolerant disjunct matrix. Discret. Appl. Math. 157(2), 387–390 (2009)
Y. Cheng, D.Z. Du, New constructions of one- and two-stage pooling designs. J. Comput. Biol. 15, 195–205 (2008)
Y. Cheng, K.-I Ko, W. Wu, On the complexity of non-unique probe selection. Theor. Comput. Sci. 390(1), 120–125 (2008)
Y. Cheng, D.Z. Du, K.-I. Ko, G. Lin, On the parameterized complexity of pooling design. J. Comput. Biol. 16, 1529–1537 (2009)
Y. Cheng, D.Z. Du, G. Lin, On the upper bounds of the minimum number of rows of disjunct matrices. Optim. Lett. 3(2), 297–302 (2009)
A. De Bonis, L. Gasieniec, U. Vaccaro, Optimal two-stage algorithms for group testing problems. SIAM J. Comput. 34, 1253–1270 (2005)
R. Dorfman, The detection of defective members of large populations. Ann. Math. Stat. 14, 436–440 (1943)
R.G. Downey, M.R. Fellows, Parameterized Complexity (Springer, New York, 1999)
D.-Z. Du, F.K. Hwang, Pooling Designs and Nonadaptive Group Testing: Important Tools for DNA Sequencing (World Scientific, New Jersey, 2006)
D.-Z. Du, K.-I. Ko, Some completeness results on decision trees and group testing. SIAM J. Algebra. Discret. 8(4), 762–777 (1987)
D.-Z. Du, K.-I. Ko, Theory of Computational Complexity (Wiley, New York, 2000)
D.Z. Du, F.K. Hwang, W. Wu, T. Znati, New construction for transversal design. J. Comput. Biol. 13, 990–995 (2006)
A.G. D’yachkov, V.V. Rykov, Bounds of the length of disjunct codes. Probl. Control Inf. Theory 11, 7–13 (1982)
A.G. D’yachkov, V.V. Rykov, A.M. Rashad, Superimposed distance codes. Probl. Control Inf. Theory 18, 237–250 (1989)
A.G. D’yachkov, A.J. Macula, V.V. Rykov, New constructions of superimposed codes. IEEE Trans. Inf. Theory 46, 284–290 (2000)
D. Eppstein, M.T. Goodrich, D.S. Hirschberg, Improved combinatorial group testing algorithms for real-world problem sizes. SIAM J. Comput. 36, 1360–1375 (2007)
P. Erdős, P. Frankl, Z. Füredi, Families of finite sets in which no set is covered by the union of r others. Isr. J. Math. 51, 79–89 (1985)
M. Farach, S. Kannan, E. Knill, S. Muthukrishnan, Group testing problems with sequences in experimental molecular biology, in Proceedings of the Compression and Complexity of Sequences, ed. by B. Carpentieri et al. (IEEE Press, Los Alamitos, 1997), pp. 357–367
J. Flum, M. Grohe, Parameterized Complexity Theory. Texts in Theoretical Computer Science, an EATCS Series, vol. XIV (Springer, Berlin, 2006)
H.L. Fu, F.K. Hwang, A novel use of t-packings to construct d-disjunct matrices. Discret. Appl. Math. 154, 1759–1762 (2006)
Z. Füredi, On r-cover-free families. J. Comb. Theory Ser. A 73, 172–173 (1996)
M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, San Francisco, 1979)
R. Herwig, A.O. Schmitt, M. Steinfath, J. O’ Brien et al., Information theoretical probe selection for hybridisation experiments. Bioinformatics 16, 890–898 (2000)
F.K. Hwang, V.T. Sós, Non-adaptive hypergeometric group testing. Studia Sci. Math. Hung. 22, 257–263 (1987)
P. Indyk, H.Q. Ngo, A. Rudra, Efficiently decodable non-adaptive group testing, in Proceedings of 21st Annual ACM-SIAM Symposium on Discrete Algorithms, Austin, 2010, pp. 1126–1142
W.H. Kautz, R.C. Singleton, Nonrandom binary superimposed codes. IEEE Trans. Inf. Theory 10, 363–377 (1964)
G.W. Klau, S. Rahmann, A. Schliep, M. Vingron, K. Reinert, Optimal robust non-unique probe selection using integer linear programming. Bioinformatics 20, i186–i193 (2004)
E. Knill, Lower bounds for identifying subset members with subset queries, in Proceedings of 6th ACM-SIAM Symposium on Discrete Algorithms, San Francisco, CA, USA, 1995, pp. 369–377
O. Lichtenstein, A. Pnueli, Checking that finite state concurrent programs satisfy their linear specification, in Proceedings of 12th ACM Symposium on Principles of Programming Languages (POPL’ 85), 107, New York, 1985, pp. 97–107
A.J. Macula, A simple construction of d-disjunct matrices with certain constant weights. Discret. Math. 162, 311–312 (1996)
A.J. Macula, Error-correcting nonadaptive group testing with d e-disjunct matrices. Discret. Appl. Math. 80, 217–222 (1997)
A.J. Macula, Probabilistic nonadaptive group testing in the presence of errors and DNA library screening. Ann. Comb. 3, 61–69 (1999)
A.J. Macula, Probabilistic nonadaptive and two-stage group testing with relatively small pools and DNA library screening. J. Comb. Optim. 2, 385–397 (1999)
R. Motwani, P. Raghavan, Randomized Algorithms (Cambridge University Press, New York, 1995)
H.Q. Ngo, D.Z. Du, A survey on combinatorial group testing algorithms with applications to DNA library screening, in DIMACS: Series in Discrete Mathematics and Theoretical Computer Science, vol. 55 (American Mathematical Society, Providence, 2000), pp. 171–182
H.Q. Ngo, D.Z. Du, New constructions of non-adaptive and error-tolerance pooling designs. Discret. Math. 243, 161–170 (2002)
H.Q. Ngo, E. Porat, A. Rudra, Efficiently decodable error-correcting list disjunct matrices and applications, in Proceedings of the 38th International Colloquium on Automata, Languages and Programming, Zurich, Switzerland, 2011, pp. 557–568
M. Olson, L. Hood, C. Cantor, D. Botstein, A common language for physical mapping of the human genome. Science 245, 1434–1435 (1989)
C.H. Papadimitriou, Computational Complexity (Addison-Wesley, New York, 1994)
C.H. Papadimitriou, D. Wolfe, The complexity of facets resolved, J. Comput. Syst. Sci. 37, 2–13 (1988)
H. Park, W. Wu, Z. Liu, X. Wu, H.G. Zhao, DNA screening, pooling design and simplicial complex. J. Comb. Optim. 7, 389–394 (2003)
E. Porat, A. Rothschild, Explicit non-adaptive combinatorial group testing schemes, in Proceedings of the 35th International Colloquium on Automata, Languages and Programming, Reykjavik, Iceland, 2008, pp. 748–759
S. Rahmann, Rapid large-scale oligonucleotide selection for microarrays, in Proceedings of the 1st IEEE Computer Society Conference on Bioinformatics (CSB’ 02), Stanford, CA, USA, 2002, pp. 54–63
S. Rahmann, Fast and sensitive probe selection for DNA chips using jumps in matching statistics, in Proceedings of the 2nd IEEE Computer Society Bioinformatics Conference (CSB’ 03), Stanford, CA, USA, 2003, pp. 57–64
M. Ruszinkó, On the upper bound of the size of the r-cover-free families. J. Comb. Theory Ser. A 66, 302–310 (1994)
A. Schliep, D.C. Torney, S. Rahmann, Group testing with DNA chips: generating designs and decoding experiments, in Proceedings of the 2nd IEEE Computer Society Bioinformatics Conference (CSB’ 03), Stanford, CA, USA, 2003, pp. 84–93
C. Umans, The minimum equivalent DNF problem and shortest implicants, in Proceedings of 39th IEEE Symposium on Foundation of Computer Science, Palo Alto, CA, USA, 1998, pp. 556–563
X. Wang, B. Seed, Selection of oligonucleotide probes for protein coding sequences. Bioinformatics 19, 796–802 (2003)
J.K. Wolf, Born again group testing: multiaccess communications. IEEE Trans. Inf. Theory IT-31, 185–191 (1985)
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Cheng, Y. (2013). Advances in Group Testing. In: Pardalos, P., Du, DZ., Graham, R. (eds) Handbook of Combinatorial Optimization. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7997-1_71
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DOI: https://doi.org/10.1007/978-1-4419-7997-1_71
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