Approximating the Minimum Equivalent Digraph

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.Comment: conference version in ACM-SIAM Symposium on Discrete Algorithms (1994

Transitive compaction in parallel via branchings

We study the following problem: given a strongly connected digraph, find a minimal strongly connected spanning subgraph of it. Our main result is a parallel algorithm for this problem, which runs in polylog parallel time and uses O(n 3) processors on a PRAM. Our algorithm is simple and the major tool it uses is computing a minimum-weight branching with zero-one weights. We also present sequential algorithms for the problem that run in time O(m + n ⋅ log n)