On Strongly Connected Digraphs with Bounded Cycle Length

TL;DR

This paper proves that the minimum equivalent graph problem is polynomial-time solvable for graphs with cycles of length at most three, improving approximation guarantees for the general case.

Contribution

It establishes the equivalence of the MEG problem to maximum bipartite matching for graphs with bounded cycle length, enabling efficient solutions.

Findings

MEG problem is polynomial-time solvable for graphs with cycles of length ≤ 3.

Improves approximation algorithms for the general MEG problem.

Provides a new characterization linking cycle length constraints to bipartite matching.

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 a proof that, for graphs where each directed cycle has at most three edges, the MEG problem is equivalent to maximum bipartite matching, and therefore solvable in polynomial time. This leads to an improvement in the performance guarantee of the previously best approximation algorithm for the general problem in ``Approximating the Minimum Equivalent Digraph'' (1995).