Approximating the Minimum Equivalent Digraph

TL;DR

This paper presents an approximation algorithm for the NP-hard minimum equivalent digraph problem, achieving a performance guarantee of approximately 1.64 by contracting long cycles, and also analyzes a local improvement algorithm with a guarantee of 1.75.

Contribution

It introduces a novel approximation algorithm with a proven performance guarantee for the minimum equivalent digraph problem, improving upon previous bounds.

Findings

Approximation algorithm with guarantee of ~1.64

Analysis of 2-Exchange local improvement algorithm with guarantee of 1.75

Method based on contracting long cycles

Abstract

The MEG (minimum equivalent graph) problem is, given a directed graph, to find a small subset of the edges that maintains all reachability relations between nodes. The problem is NP-hard. This paper gives an approximation algorithm with performance guarantee of pi^2/6 ~ 1.64. The algorithm and its analysis are based on the simple idea of contracting long cycles. (This result is strengthened slightly in ``On strongly connected digraphs with bounded cycle length'' (1996).) The analysis applies directly to 2-Exchange, a simple ``local improvement'' algorithm, showing that its performance guarantee is 1.75.