Faster and Simpler Minimal Conflicting Set Identification

TL;DR

This paper introduces a faster, simpler algorithm for identifying minimal conflicting sets related to the Consecutive Ones Property, significantly improving computational efficiency by leveraging interval graph theory.

Contribution

The authors present a novel algorithm that reduces complexity for MCS detection by using minimal forbidden induced subgraphs, surpassing previous methods in speed and simplicity.

Findings

Algorithm runs in O(N^2 M^2 + NM^7) time, faster than previous methods.

Uses interval graph theory instead of Tucker matrices for MCS detection.

Significantly improves efficiency in identifying conflicting sets.

Abstract

Let C be a finite set of N elements and R = r_1,r_2,..., r_m a family of M subsets of C. A subset X of R verifies the Consecutive Ones Property (C1P) if there exists a permutation P of C such that each r_i in X is an interval of P. A Minimal Conflicting Set (MCS) S is a subset of R that does not verify 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 R belongs to at least one MCS. Our algorithm runs in O(N^2M^2 + NM^7), largely improving the current O(M^6N^5 (M+N)^2 log(M+N)) fastest algorithm of [Blin {\em 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.