FasterDSP: A Faster Approximation Algorithm for Directed Steiner Tree Problem *
Ming I. Hsieh, Eric Hsiao Kuang Wu, Wu, Meng‐Feng Tsai · 2008
Given a weighted directed graph G = (V, E, c), where c: E → R + is an edge cost function, a subset X of vertices (terminals), and a root vertex vr, the directed Steiner tree problem (DSP) asks for a minimum-cost tree which spans the paths from root vertex vr to each terminal. Charikar et al.’s algorithm is well-known for this problem. It achieves 1 an approximation guarantee of ( 1) l ll − k in O(n l k 2l) time for any fixed level l> 1, where l is the level of the tree produced by the algorithm, n is the number of vertices, |V|, and k is the number of terminals, |X|. However, it requires a great amount of computing power, and there are some problems in the proof of the approximation guarantee of the algorithm. This paper provides a faster approximation algorithm improving Charikar et al.’s DSP algorithm with a better time complexity, O(n l k l + n 2 k + nm), where m is the number of edges, and an amended 8k − δ lnk factor for the 2-level Steiner tree, where δ =