Consecutive ones property and spectral ordering

Abstract

A 0–1 matrix where in each column the 1s occur consecutively is said to have the consecutive ones property. This property and its approximations are important in a variety of applications, e.g., in DNA analysis and paleontology. Checking whether the matrix rows can be permuted so that this property holds can be done in polynomial time. However, an exact solution rarely exists in applications. Thus, methods that produce good approximations with a minimal number of 0s between 1s in the columns are needed. The spectral ordering method gives a solution to the consecutive ones problem and has empirically been shown to work well in other cases too. This paper constructs a theoretical basis for the connection between the consecutive ones property and spectral ordering. We represent the approximation problem as a problem of minimizing a certain matrix norm. We show how the spectral ordering method affects this norm and show that the method is optimal within a class of algorithms. We analyze also popular normalization schemes for spectral ordering and give our recommendation on their use for the C1P problem. We use these theoretical results to explain the experimental performance of different approximate methods and their differences.