Faster and Simpler Minimal Conflicting Set Identification

Abstract

Let ${\cal C}$ be a finite set of n elements and ${\cal R}=\{r_1,r_2, \ldots , r_m\}$ a family of m subsets of ${\cal C}$. A subset ${\cal X}$ of ${\cal R}$ satisfies the Consecutive Ones Property (C1P) if there exists a permutation P of ${\cal C}$ such that each r _i in ${\cal X}$ is an interval of P. A Minimal Conflicting Set (MCS) ${\cal S} \subseteq{\cal R}$ is a subset of ${\cal R}$ that does not satisfy the C1P, but such that any of its proper subsets does. In this paper, we present a new simpler and faster algorithm to decide if a given element $r \in{\cal R}$ belongs to at least one MCS. Our algorithm runs in O(n ² m ² + nm ⁷), largely improving the current O(m ⁶ n ⁵ (m + n)² log(m + n)) fastest algorithm of [Blin et al, CSR 2011]. The new algorithm is based on an alternative approach considering minimal forbidden induced subgraphs of interval graphs instead of Tucker matrices.