$d$-COS-R is FPT via Interval Deletion

TL;DR

This paper proves that the problem of deleting at most d rows to achieve the Consecutive Ones Property in a binary matrix is fixed-parameter tractable, with an algorithm running in time O^*(10^d).

Contribution

It introduces a recursive search tree algorithm that reduces the problem to Interval Deletion, establishing fixed-parameter tractability for d-COS-R.

Findings

d-COS-R is fixed-parameter tractable (FPT).

The algorithm runs in time O^*(10^d).

The approach reduces to solving Interval Deletion instances.

Abstract

A binary matrix $M$ has the Consecutive Ones Property (COP) if there exists a permutation of columns that arranges the ones consecutively in all the rows. Given a matrix, the $d$-COS-R problem is to determine if there exists a set of at most $d$ rows whose deletion results in a matrix with COP. We consider the parameterized complexity of this problem with respect to the number $d$ of rows to be deleted as the parameter. The closely related Interval Deletion problem has recently shown to be FPT [Y. Cao and D. Marx, Interval Deletion is Fixed-Parameter Tractable, arXiv:1211.5933 [cs.DS],2012]. In this work, we describe a recursive depth-bounded search tree algorithm in which the problems at the leaf-level are solved as instances of Interval Deletion. The running time of the algorithm is dominated by the running time of Interval Deletion, and therefore we show that $d$-COS-R is fixed-parameter tractable and has a run-time of $O^*(10^d)$.