Motivation and Introduction

  1. What is Group Testing? Applications. Problem formulation.

    Motivations for the group testing problem are discussed. Examples include blood testing in medicine, traitor tracing and broadcast encryption, live baiting DoS attackers and data forensics in security, drug and DNA library screening in computational biology, MAC protocol in networking, compressed sensing in signal processing, heavy hitter in data streaming. We will discuss a connection to sparse recovery. Then, notions of disjunct and separable matrices are discussed, along with basic relations and bounds. Objectives of group testing are outlined: reducing the number of tests, improving time and space complexity, explicit and strongly explicit constructions, various constraints on the tests (arithmetic progressions, bounds on test sizes), error-tolerance, list group testing and MUT.

  2. Existence proof. Basic probabilistic and algorithmic construction. Basic lower bounds

    Lower bounds: the basic Bassalygo's bound and Erdos-Frankl-Furedi's technique. Upper bounds: probabilistic existence, Kautz-Singleton code concatenation technique with MDS codes, RS construction, greedy algorithm with set cover analysis (maybe also Hwang-Sos construction).

Computational complexity and algorithmic aspects. Error-free case.

  1. Basic complexity results and approximation algorithms.

    Hardness of group testing. We might have to skip this lecture in the interest of time.

  2. Efficient constructions of efficiently decodable non-adaptive group testing strategies.

    Porat-Rothschild's construction. Indyk-Ngo-Rudra's code concatenation idea. Ngo-Porat-Rudra's two constructions, one based on code concatenation, the other based on recursion. Show how they are efficiently decodable. Coding theory and list-recoverable codes.

Error-tolerant case.

  1. Bounds for the error-tolerant case.

    We will prove analogous lower and probabilistic upper bounds for the error-correcting case.

  2. Efficient constructions of efficiently decodable and error-tolerant non-adaptive group testing strategies.

    We describe how the Ngo-Porat-Rudra's constructions can be made error-tolerant.

Extensions and open problems. (We won't have time to cover)

  1. Variations of the basic group testing paradigm. Connection to compressive sensing.

    There are many variations of the group testing problem. We will discuss some of them here, including tests with bounded size (upper, lower), threshold group testing, adder channel and compressive sensing, arithmetic progressiong tests, and transversal group testing.

  2. Fundamental open problems.

    We explicitly list open problems which are implicit in the above.