A note on a 2-approximation algorithm for the MRCT problem∗

Consider the following problem in network design: given an undirected graph with nonnegative delays on the edges, the goal is to find a spanning tree such that the average delay of communicating between any pair using the tree is minimized. The delay between a pair of vertices is the sum of the delays of the edges in the path between them in the tree. Minimizing the average delay is equivalent to minimizing the total delay between all pairs of vertices using the tree. In general, when the cost on an edge represents a price for routing messages between its endpoints (such as the delay), the routing cost for a pair of vertices in a given spanning tree is defined as the sum of the costs of the edges in the unique tree path between them. The routing cost of the tree itself is the sum over all pairs of vertices of the routing cost for the pair in this tree, i.e., C(T ) = ∑ u,v dT (u, v), where dT (u, v) is the distance between u and v on T . For an undirected graph, the minimum routing cost spanning tree (MRCT) is the one with minimum routing cost among all possible spanning trees. Unless specified explicitly in this note, a graph G is assumed to be simple and undirected, and the edge weights are nonnegative. Finding an MRCT in a general edge-weighted undirected graph is known to be NP-hard. In this note, we shall focus on the 2-approximation algorithms. Before going into the details, we introduce a term, routing load, which provides us an alternative formula to compute the routing cost of a tree.