Title : 
  Group Testing Designs :
A Combinatorial Marvel
Speaker: Bikas K Sinha

Retired Professor of Statistics

Indian Statistical Institute, Kolkata

Abstract :

Group Testing is a technique to test a collection of units in several 
groups, rather than in isolation (i.e., one-at-a-time), in order to 
ascertain the 'status' of each individual unit in the collection in 
respect of a well-defined 'feature'. The problem is to plan the testing 
procedure so as to be able to do so without any ambiguity and with a 
minimum number of such tests [called Group Tests (GTs)]. The response to 
be extracted from each unit is on the same 'feature' and it is 'binary' 
in nature.  It is tacitly assumed that 'possession' of the feature by at 
least one unit within a group [so formed] would render the group 
'identifiable' as 'possessed'. When this happens, we need to 'open up' 
the group and go for further exploration of the status of individual 
units of the group, possibly by sub-group(s) testing or by other means. 
Other possibility is that the group would be declared as 'passed', and 
consequently, it would mean that all constituent units within the group 
would be declared as 'passed' and 'at one go'!

This interpretation is accepted for Group Testing schemes to work. When 
this latter phenomenon happens, the merit of Group Testing prevails over 
testing individual units in terms of reduction in the required number of 
tests. For a given collection of units, we may adopt one-at-a-time 
testing or Group Testing with formation of suitable groups, or even a 
combination of the two strategies. As is mentioned above, the sole 
purpose is to minimize the number of GTs in such situations for a given 
collection of test items.

The above formulation looks deceptively simple! Hidden are probabilistic 
and combinatorial challenges. In this talk, we will discuss some issues 
related to combinatorial challenges only.

Key Words.... Group tests; Hypergeometric group tests, Sequential group 
tests, t-completeness, Detecting power of order t, Group Divisible 
Designs, Petersen graphs.