APPROXIMATING THE MINIMUM EQUIVALENT DIGRAPH

The minimum equivalent graph (MEG) problem is as follows: given a directed graph, find a smallest subset of the edges that maintains all reachability relations between nodes. This problem is NP-hard; this paper gives an approximation algorithm achieving a performance guarantee of about 1.64 in polynomial time. 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 strongly connected spanning subgraph (SCSS) 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.