Approximation Algorithms for Minimum-Time Broadcast under the Vertex-Disjoint Paths Mode

We give a polynomial-time O(log n/log OPT)-approximation algorithm for minimum-time broadcast and minimum-time multicast in n-node networks under the single-port vertex-disjoint paths mode. This improves a previous approximation algorithm by Kortsarz and Peleg. In contrast, we give an Ω(log n) lower bound for the approximation ratio of the minimum-time multicast problem in directed networks. This lower bound holds unless NP ⊂ DTIME(nlog log n). An important consequence of this latter result is that the Steiner version of the Minimum Degree Spanning Tree (MDST) problem in digraphs cannot be approximated within a constant ratio, as opposed to the undirected version. Finally, we give a polynomial-time O(1)-approximation algorithm for minimum-time gossip (i.e., all-to-all broadcast).