A Practical Greedy Approximation for the Directed Steiner Tree Problem

Abstract

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 $O(n m k^2)$.

We provide computational results to show that, Greedy$_\mathrm{FLAC }$ rivals the running time of the fast $k$-approximation and returns solution with smaller cost in practice.