Maximal and maximum transitive relation contained in a given binary relation

TL;DR

This paper investigates algorithms for finding maximal and maximum transitive relations within a binary relation or digraph, providing polynomial-time solutions and approximation algorithms with bounds for specific graph classes.

Contribution

It introduces a polynomial-time algorithm for maximal transitive sub-relations and a 0.874-approximation algorithm for maximum transitive subgraphs in triangle-free digraphs.

Findings

Polynomial-time algorithm for maximal transitive sub-relation.

0.874-approximation algorithm for maximum transitive subgraph in triangle-free digraphs.

Upper bound on maximum transitive relation size in terms of edges.

Abstract

We study the problem of finding a \textit{maximal} transitive relation contained in a given binary relation. Given a binary relation of size $m$ defined on a set of size $n$, we present a polynomial time algorithm that finds a maximal transitive sub-relation in time $O(n^2 + nm)$. We also study the problem of finding a \textit{maximum} transitive relation contained in a binary relation. This is the problem of computing a maximum transitive subgraph in a given digraph. For the class of directed graphs with the underlying graph being triangle-free, we present a $0.874$-approximation algorithm. This is achieved via a simple connection to the problem of maximum directed cut. Further, we give an upper bound for the size of any maximum transitive relation to be $m/4 + cm^{4/5}$, where $c > 0$ and $m$ is the number of edges in the digraph.