Efficient Algorithms for Finding Tucker Patterns

TL;DR

This paper presents linear-time algorithms for detecting Tucker patterns in binary matrices that lack the Consecutive Ones Property, enabling efficient certification and enumeration of these forbidden submatrices.

Contribution

It introduces a linear-time algorithm for finding Tucker patterns and an output-sensitive method for enumerating all such patterns in non-C1P matrices.

Findings

Linear-time algorithm for Tucker pattern detection

Output-sensitive enumeration of Tucker patterns

Efficient certification of non-C1P matrices

Abstract

The Consecutive Ones Property is an important notion for binary matrices, both from a theoretical and applied point of view. Tucker gave in 1972 a characterization of matrices that do not satisfy the Consecutive Ones Property in terms of forbidden submatrices, the Tucker patterns. We describe here a linear time algorithm to find a Tucker pattern in a non-C1P binary matrix, which allows to extract in linear time a certificate for the non-C1P. We also describe an output-sensitive algorithm to enumerate all Tucker patterns of a non-C1P binary matrix. This paper had been withdrawn due to some missing cases in Algorithms 2 and 3.