Improved Approximations for the Hotlink Assignment Problem

Let G = (V E) be a graph representing a Web site, where nodes correspond to pages and arcs to hyperlinks. In this context, hotlinks are defined as shortcuts (new arcs) added to Web pages of G in order to reduce the time spent by users to reach their desired information. In this article, we consider the problem where G is a rooted directed tree and the goal is minimizing the expected time spent by users by assigning at most k hotlinks to each node. For the most studied version of this problem where at most one hotlink can be added to each node, we prove the existence of two FPTAS's which optimize different objectives considered in the literature: one minimizes the expected user path length and the other maximizes the expected reduction in user path lengths. These results improve over a constant factor approximation for the expected length and over a PTAS for the expected reduction, both obtained recently in Jacobs [2007]. Indeed, these FPTAS's are essentially the best possible results one can achieve under the assumption that P not equal NP. Another contribution we give here is a 16-approximation algorithm for the most general version of the problem where up to k hotlinks can be assigned from each node. This algorithm runs in O(vertical bar V vertical bar log vertical bar V vertical bar) time and it turns to be the first algorithm with constant approximation for this problem.