Approximating the minimum equivalent digraph

The MEG (minimum equivalent graph) problem is the following: `Given a directed graph, find a smallest subset of the edges that maintains all reachability relations between nodes.' The MEG problem is NP-hard; this paper gives an approximation algorithm achieving a performance guarantee of about 1.64 in polynomial time. We give a modification that improves the performance guarantee to about 1.61. The algorithm achieves a performance guarantee of 1.75 in the time required for transitive closure. The heart of the MEG problem is the minimum SCSS (strongly connected spanning subgraph) problem - the MEG problem restricted to strongly connected digraphs. For the minimum SCSS problem, the paper gives a practical, nearly linear-time implementation achieving a performance guarantee of 1.75. The algorithm and its analysis are based on the simple idea of contracting long cycles. The analysis applies directly to 2-EXCHANGE, a general `local improvement' algorithm, showing that its performance guarantee is 1.75.