Motivations and some Background Materials

  1. [ pdf ] What is Group Testing? Problem formulation, prototypical applications, the matrix representations with disjunct and separable matrices. Four main objectives: (1) reducing number of tests, (2) efficient constructions, (3) efficient decoding, (4) error-tolerance.
  2. [ pdf ] Lower bounds on the number of tests.
    • M. Ruszinko, On the upper bound of the size of the r-cover-free families, J. Combin. Theory Ser. A, 66 (1994), pp. 302–310. [ pdf ]
    • Z. Furedi, On r-cover-free families, Journal of Combinatorial Theory, Ser. A 73 (1996), 172-173. [ pdf ]
    • P. Erdös, P. Frankl, and Z. Furedi: Families of finite sets in which no set is covered by the union of r others, Israel Journal of Mathematics 51 (1985), 79-89. [ pdf ]
  3. [ pdf ] Brief introduction to coding theory: codes, Singleton bound, linear codes, Reed-Solomon codes, code concatenation, Gilbert-Varshamov bound.
  4. [ pdf ] Probabilistic upper bounds on the number of tests. The code concatenation technique. Concatenating a random code with the identity code. Connection to the k-restriction problem.
    • Even, G., Goldreich, O., Luby, M., Nisan, N. and Veličković, B. (1998), Efficient approximation of product distributions . Random Structures and Algorithms, 13: 1–16. [ pdf ]
    • Noga Alon, Dana Moshkovitz, and Shmuel Safra. 2006. Algorithmic construction of sets for k-restrictions. ACM Trans. Algorithms 2, 2 (April 2006), 153-177.

Constructions of group testing matrices. Error-free case.

  1. [ pdf ] Concatenating the RS-code with the identity code. A greedy construction. The Porat-Rothschild construction.
    • E. Porat, A. Rothschild, "Explicit Non-Adaptive Combinatorial Group Testing Schemes", ICALP'08.
    • Sergey Yekhanin, Some new Constructions of Optimal Superimposed Designs [ pdf ]
    • Arkadii G. D'yachkov, Anthony J. Macula, Vyacheslav V. Rykov: New constructions of superimposed codes. IEEE Transactions on Information Theory 46(1): 284-290 (2000)
    • David Eppstein, Michael T. Goodrich, Daniel S. Hirschberg: Improved Combinatorial Group Testing Algorithms for Real-World Problem Sizes. SIAM J. Comput. 36(5): 1360-1375 (2007)
  2. Basic complexity results and approximation algorithms.
    • [ pdf ] Du, Ding Zhu; Ko, Ker-I Some completeness results on decision trees and group testing. SIAM J. Algebraic Discrete Methods 8 (1987), no. 4, 762–777.
    • [ pdf ] Feng(PRC-ZHJ); Du, Ding Zhu(PRC-ASBJ-AM) The complexity of determinacy problem on group testing. Discrete Appl. Math. 28 (1990), no. 1, 71–81.

Constructions of efficiently decodable group testing matrices. Error-free case.

  1. [ pdf ] Efficient decoding and list group testing
  2. [ pdf ] Basic bounds for the number of tests of list disjunct and list separable matrices
  3. [ pdf ] A recursive construction of sub-linear time decodable list disjunct matrices
    • Hung Q. Ngo, Ely Porat, Atri Rudra: Efficiently Decodable Error-Correcting List Disjunct Matrices and Applications - (Extended Abstract). ICALP (1) 2011: 557-568
  4. [ pdf ] Constructions of list disjunct matrices based on list-recoverable codes.
    • Piotr Indyk, Hung Q. Ngo, Atri Rudra: Efficiently Decodable Non-adaptive Group Testing. SODA 2010: 1126-1142
  5. [ pdf ] Explicit construction of list disjunct matrices based on randomness extractors and based on expanders.
    • Mahdi Cheraghchi: Noise-Resilient Group Testing: Limitations and Constructions. FCT 2009: 62-73

Error-tolerant case.

  1. [ pdf ] Error-correcting (list-) separable and (list-) disjunct matrices. Lower bounds and probabilistic upper bounds.
  2. Explicit and strongly explicit constructions of error-correcting disjunct matrices
  3. Explicit and strongly explicit constructions of efficiently decodable and error-correcting list-disjunct matrices
  4. The relative error rate case.

Extensions, applications.

  • Variations of the basic group testing paradigm.
    • Mahdi Cheraghchi: Improved Constructions for Non-adaptive Threshold Group Testing. ICALP (1) 2010: 552-564
  • Group testing on graphs
    • Mahdi Cheraghchi, Amin Karbasi, Soheil Mohajer, Venkatesh Saligrama: Graph-Constrained Group Testing CoRR abs/1001.1445: (2010)
    • Nicholas J. A. Harvey, Mihai Patrascu, Yonggang Wen, Sergey Yekhanin, Vincent W. S. Chan: Non-Adaptive Fault Diagnosis for All-Optical Networks via Combinatorial Group Testing on Graphs. INFOCOM 2007: 697-705
    • M. Wang, W. Xu, E. Mallada and A. Tang. Sparse Recovery with Graph Constraints: Fundamental Limits and Measurement Construction. to appear in IEEE Infocom 2012.
    • W. Xu, E. Mallada and A. Tang. Compressive Sensing over Graphs. Proceedings of IEEE Infocom 2011.
  • Three applications of list-disjunct matrices.
    • Noga Alon, Rani Hod: Optimal Monotone Encodings. IEEE Transactions on Information Theory 55(3): 1343-1353 (2009)
    • N. Alon and V. Asodi, Tracing many users with almost no rate penalty, IEEE Transactions on Information Theory 53 (2007), 437-439.
    • Ravi Kumar, Sridhar Rajagopalan, Amit Sahai: Coding Constructions for Blacklisting Problems without Computational Assumptions. CRYPTO 1999: 609-623
  • Connection to compressive sensing.
    • Hung Q. Ngo, Ely Porat, and Atri Rudra, "Efficiently Decodable Compressed Sensing by List-Recoverable Codes and Recursion," in Proceedings of the 29th Symposium on Theoretical Aspects of Computer Science (STACS 2012), Feb 29 –- March 3rd, 2012, Paris, France.
  • Applications in pattern matching.
  • Applications in cryptography.
  • Applications in streaming.