A Polynomial-Time Algorithm for Finding a Minimal Conflicting Set Containing a Given Row

A binary matrix has the Consecutive Ones Property (C1P) if there exists a permutation of its columns (i.e. a sequence of column swappings) such that in the resulting matrix the 1s are consecutive in every row. A Minimal Conflicting Set (MCS) of rows is a set of rows R that does not have the C1P, but such that any proper subset of R has the C1P. In [5], Chauve et al. gave a O(∆m(n+m+ e)) time algorithm to decide if a row of a m × n binary matrix with at most ∆ 1s per row belongs to at least one MCS of rows. Answering a question raised in [2], [5] and [25], we present the first polynomial-time algorithm to decide if a row of a m× n binary matrix belongs to at least one MCS of rows.