# Study of Three Combinatorial Problems.

December 1993

## Date of Award

Winter 12-12-1994

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Master's Dissertation

## Degree Name

Master of Technology

Computer Science

## Department

Applied Statistics Unit (ASU-Kolkata)

## Supervisor

Roy, Bimal Kumar (ASU-Kolkata; ISI)

## Abstract (Summary of the Work)

The problem is to construct a set of n tests, to test a set of v items. The itens can be combined, and a test applied on the combinat- ion. A test reports positive if any of the items in the conbination is defective. If all the itens which were combined together are good then a test conducted on the combination gives a negative result. The set up is in the sense that all the tests are assumed to be non-adaptive, conducted sinultaneously, and the results of all the tests are used to discover the defective itens. The pur pose is to winimise the number of tests required, as it is aluays possible to individually test the itens requiring v tests.A set of tests to achieve the above pur pose is called a design. Such designs have been considered in t13,t2). In C13 a property of a design called detecting power is defined. A design is said to have det- t, or DP(t) if it can correctly detect the presence of ecting power upto t defective items. In the following we consider the problen for designs having DP(2). In (23, it is shown that mininizing the number of tests is equivalent to naximising the number of rous of a binary natrix with n colunns, having a certain property.Here we take an algoritheic approach to the problem. The naive algorithn is outlined, and is shown to be inefficient. The algorithm has been inplenented, and becomes impractical to run even for 7 colunns. A bound on of rows is established. A sufficient the aximua number condition for a binary satrix to have DP(2) is used to get a heuristic for the problem. A counting argunent is used to get a result which puts a bound on the number of rous that can be included in a matrix, when two other rows are present.

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843046

ISI-DISS-1993-