On the Complexity of Optimizing Pagerank

EXTENDED ABSTRACT. In search engines, it is critical to be able to compare webpages according to their relative importance, with as few as possible computational resources. This is done by computing the PageRank of every webpage from the web [4] (i.e., the average portion of time spent in the webpage during an infinite and uniform random walk on the web). Pages with higher PageRank will then appear higher in the list of results. See [3] for a survey on PageRank and its applications. The concept of PageRank has generated a large number of questions and challenges. Among these, the problem of optimizing the PageRank of webpages raises increasing interest, as evidenced by the growing literature on the subject [1, 20, 7, 5, 17, 6, 10]. In the PageRank Optimization problem (PRO) that we study, one aims to maximize (or minimize) the PageRank of some target node when control is granted on some subset of free edges that may be chosen to be activated or deactivated. A typical application of PRO is the so-called webmaster problem in which a webmaster tries to maximize the PageRank of one of his webpage by determining which links under his control (i.e. on his website, or on an allied website for instance) he should activate and which links he should not [1, 7]. The same tools may be used to find how much the PageRank of some nodes can vary when the presence or absence of some links is uncertain (e.g. because a link is broken, the server is down or because of traffic problems) [17]. Similar situations also exist in economic networks in which agents choose partners in order to increase their centrality (i.e., their PageRank) [22] or decrease the centrality of other agents. For instance, a government might want to prevent a bank from acquiring excessive influence in order to limit the sensitivity of the bank network to a possible bankruptcy, and it should therefore allow or reject some transactions [2, 8, 11, 19]. Csáji et al. proposed a way of modeling PRO as a Markov Decision Process (MDP), thereby showing that an exact solution of the problem could be found in weakly polynomial time using linear programming [6]. (For more on MDP see, e.g., [21].) Yet in practice, MDPs are solved much more efficiently using algorithms adapted to their special structure. Among these algorithms, Policy Iteration (PI) [16] performs very well in practice and is guaranteed to converge to the optimal solution in a finite number of iterations. However, even though PI usually converges in a few iterations, its actual complexity remains unclear.