Minimising the number of gap-zeros in binary matrices

Abstract

We study a problem of minimising the total number of zeros in the gaps between blocks of consecutive ones in the columns of a binary matrix by permuting its rows. The problem is refered to as the Consecutive One Augmentation Problem, and is known to be NP-hard. An analysis of the structure of an optimal solution allows us to focus on a restricted solution space, and to use an implicit representation for searching the space. We develop an exact solution algorithm, fixed-parameter tractable with respect to the number of columns, and two constructive heuristics to tackle instances with an arbitrary number of columns. The heuristics use a novel solution representation based upon row sequencing. In our computational study, all heuristic solutions are either optimal or close to an optimum. One of the heuristics is particularly effective, especially for problems with a large number of rows.

Alleviating bottlenecks in a microbiology laboratory

“…The mathematical abstraction of the problem of sequencing agar samples relates to matrices with 0-1 entries, as explained by Chakhlevitch et al (2010a), which presents the general mathematical problem and solutions. An exact algorithm is developed which can solve problem instances arising in PAS microbiology laboratory optimally as illustrated in case study 2.…”

“…Using the algorithm developed in Chakhlevitch et al (2010a) the test suites can be permuted to the list in Table A4 containing only 10 gap-zeros (highlighted in bold) with the relatively small total weight of 68 reflecting short breaks in usage of agars. Reducing the total size of the gaps in agar consumption takes pressure off the agar production machines.…”