[Paper Review] An approximation algorithm for the link building problem

Martin Olsen, Anastasios Viglas+1 more|arXiv (Cornell University)|Apr 5, 2012

Complexity and Algorithms in Graphs7 references4 citations

TL;DR

This paper presents a polynomial-time approximation algorithm that guarantees a constant-factor approximation for maximizing the PageRank of a target node by adding k backlinks in a directed graph. Unlike a naive greedy approach that selects links from high PageRank nodes, the proposed algorithm improves reachability and PageRank more effectively by considering structural properties, achieving a theoretical approximation ratio of at least $\frac{1}{1 - \alpha^2}$, which approaches 3.6 for $\alpha = 0.85$. The algorithm is proven to be in APX, and the paper establishes a lower bound on the performance of simpler heuristics.

ABSTRACT

In this work we consider the problem of maximizing the PageRank of a given target node in a graph by adding $k$ new links. We consider the case that the new links must point to the given target node (backlinks). Previous work shows that this problem has no fully polynomial time approximation schemes unless $P=NP$. We present a polynomial time algorithm yielding a PageRank value within a constant factor from the optimal. We also consider the naive algorithm where we choose backlinks from nodes with high PageRank values compared to the outdegree and show that the naive algorithm performs much worse on certain graphs compared to the constant factor approximation scheme.

Motivation & Objective

• To develop a polynomial-time algorithm that approximates the optimal solution for maximizing PageRank of a target node by adding k backlinks.
• To analyze the performance of a naive greedy heuristic that selects links from nodes with high PageRank-to-outdegree ratio.
• To establish a theoretical lower bound on the approximation ratio achievable by such heuristics.
• To prove that the Link Building problem is in APX, meaning a constant-factor approximation is possible in polynomial time.
• To demonstrate that no fully polynomial-time approximation scheme (FPTAS) exists unless P = NP, highlighting the problem's computational hardness.

Proposed method

• The algorithm uses a reachability-based greedy strategy, iteratively selecting nodes that maximize the increase in the target node's reachability in the random surfer model.
• It leverages the recursive PageRank update formula $ f^{r+1}_i(B \cup \{y\}) \geq f^{r+1}_i(B) $, proving monotonic improvement in reachability with each added link.
• The method analyzes the PageRank vector through a system of linear equations derived from the identity $ \pi^T = \pi^T Q $, modeling the stationary distribution of the random surfer walk.
• It introduces a parameterized graph construction with three node types: shaded nodes (high in-degree, low out-degree), light nodes (high out-degree), and an isolated clique to control zapping effects.
• The algorithm's performance is evaluated by comparing its resulting PageRank to the optimal solution on a family of graphs parameterized by $ c $, with $ t_b = c $, $ t_c = c+1 $, and $ t_i = c^2 $.
• Theoretical analysis uses limits as $ c \to \infty $ to derive the asymptotic approximation ratio of $ \frac{1}{1 - \alpha^2} $.

Experimental results

Research questions

• RQ1 What is the best possible approximation ratio achievable by any polynomial-time algorithm for the link building problem?
• RQ2 How does a simple greedy heuristic—choosing backlinks from nodes with high PageRank relative to outdegree—perform compared to the optimal solution?
• RQ3 Can a constant-factor approximation algorithm be designed for the link building problem, despite its NP-hardness and lack of FPTAS?
• RQ4 What structural properties of the graph cause the naive heuristic to perform poorly, and how can they be exploited to improve approximation?
• RQ5 What is the theoretical lower bound on the approximation ratio of the r-Greedy algorithm for maximizing PageRank via backlinks?

Key findings

• The proposed algorithm achieves a PageRank value within a constant factor of the optimal solution, proving the problem is in APX.
• The approximation ratio of the r-Greedy algorithm is bounded below by $ \frac{1}{1 - \alpha^2} $, which equals approximately 3.6 when $ \alpha = 0.85 $, and this bound is tight in the limit as graph size increases.
• The naive heuristic, which selects links from nodes with high PageRank-to-outdegree ratio, can perform arbitrarily poorly on certain graph families, with an approximation ratio approaching infinity in the worst case.
• The optimal solution often benefits from adding links from nodes that create short loops (i.e., high $ z_{xx} $), which the r-Greedy algorithm fails to exploit.
• The paper proves that no fully polynomial-time approximation scheme (FPTAS) exists for the link building problem unless P = NP, establishing its computational hardness.
• The analysis shows that the performance gap between the naive heuristic and the optimal solution can be made arbitrarily large by constructing specific graph families with increasing size.