A Practical Greedy Approximation for the Directed Steiner Tree Problem

The Directed Steiner Tree (DST) NP-hard problem asks, considering a directed weighted graph with \(n\) nodes and \(m\) arcs, a node \(r\) called root and a set of \(k\) nodes \(X\) called terminals, for a minimum cost directed tree rooted at \(r\) spanning \(X\). The best known polynomial approximation ratio for DST is a \(O(k^\varepsilon )\)-approximation greedy algorithm. However, a much faster \(k\)-approximation, returning the shortest paths from \(r\) to \(X\), is generally used in practice. We give in this paper a new \(O(\sqrt{k})\)-approximation greedy algorithm called Greedy\(_\mathrm{FLAC }\) \(^\triangleright \), derived from a new fast \(k\)-approximation algorithm called Greedy\(_\mathrm{FLAC }\) running in time at most \(O(n m k^2)\).