Network Design and Game Theory Spring 2011 Lecture 6 Instructor : Mohammad T . Hajiaghayi

1 Overview During this lecture we will present approximation algorithms for three network design problems: Group Steiner Tree, k-median and k-center. Goal: Find a tree in G that connects at least one vertex from each S i. Observe that we may guess one vertex r ∈ V (e.g. by trying all vertices from S 1), that is a part of an optimum solution. We want to prove that if G is a tree rooted at r of depth h, than we can in polynomial time find a solution of cost at most O(h log p) times more than the optimum solution. The following theorem is due to Garg, Konjevod and Ravi [2], but we present a simpler proof here. Theorem 1 There is an O(h log p)-approximation algorithm for Group Steiner Tree when G = (V, E) is a tree rooted at r of depth h. Proof: For simplicity we first modify the given tree to make sure that each group S i contains only leaves of the tree G. That is if v ∈ V is an internal node and v ∈ S i , then we connect to v a leaf v by an edge of cost zero and for each group that contains v we remove v from it and add v instead.