On Physical Mapping and the consecutive ones property for sparse matrices

A first step in the investigation of long DNA molecules is often to create a library of clones which are copies of overlapping fragments of the molecule. In creating a done library the relative order of the clones along the molecule gets lost. Hence, an important problem in molecular biology is to reconstruct this order. This Physical Mapping Problem can be solved using fingerprints of the clones. A fingerprint of a clone tells, for a given set of small synthetic DNA molecules called probes, which probes bind to the clone. This data can be stored in a Boolean probe-clone incidence matrix. Of practical interest is the case when the probe-clone incidence matrix is sparse, containing only a constant number of ones in each row and column. In this paper we give a simplified model for Physical Mapping with probes that tend to occur very rarely along the DNA and show that the problem is NP-complete even for sparse matrices. Moreover, we show that Physical Mapping with chimeric clones (a clone is chimeric if it stems from a concatenation of several fragments of a DNA) is NP-complete even for sparse matrices. Both problems are modeled as variants of the Consecutive Ones Problem which makes our results interesting for other application areas.