On the approximability of the link building problem Author - MartinOlsena,AnastasiosViglasb, ∗ Speaker - Wayne Yang

Agenda Introduction Define LINK BUILDING PROBLEM Effect of receiving new links Hardness results the complexity of LINK BUILDING An approximation algorithm for LINK BUILDING Lower bounds for the approximation ratios of greedy for LINK BUILDING Discussion and Problems

Introduction Search engine optimization(SEO)is a fast growing industry that deals with optimizing the ranking of web pages in search engine results. The PageRank algorithm is one of the most well-known methods of defining a ranking among vertices according to the link structure of a graph.

LINK BUILDING PROBLEM Instance : A triple (G,x,k) where G(V,E)is a directed graph. Solution : A set S ⊆ V \{x} with |S|=k. Objective : Maximize πx in G(V,E ∪ (S×{x})). x<=target vertex k<= how many optimal link to create

Effect of receiving new links Avrachenkov and Litvak[6] study the effect on PageRank of adding newlinks with the same origin to the web graph. Theorem 1. Let each of the pages 2 to k + 1 create a link to page 1. If π˜ p denotes the updated PageRank value for page p for p ∈ {1,...,n}then we have: π ˜ p = πp + [ π2 π3... πk+1 ] M^(-1) q Roughly the first factor concerns the PageRank values of the vertices involved and the second factor M^(-1) q concerns the “distances” between the vertices involved in the update.

Ideal sources for backlinks Any vertex u in S satisfies at least one of the following two conditions: (4a) u is relatively popular compared to its out degree, or (4b) u has a low out-degree and is within a short distance from x = 1 (zxu is large) (4c) The vertices belong to different communities(zuv is small for u, v ∈ S) (4d)The distances from the vertices to x = 1 are long(zux is small for u ∈ S)

Hardness results the complexity of LINK BUILDING We show that LINKBUILDING is W[1]-hard, and does not have a fully polynomial-time approximation scheme(FPTAS). These results are based on reductions from a variant of independent set. =>If NP!=P then there is no FPTAS for LINK BUILDING.

An approximation algorithm for LINK BUILDING r-Greedy, a greedy polynomial time algorithm for LINK BUILDING computing a set of knew backlinks to a target vertex x to achieve a PageRank value that is within a constant factor from the optimal value. zuv denotes the expected number of visits to vertex v,for the PageRank random walk, starting from vertex u, before a zapping event occurs.

An approximation algorithm for LINK BUILDING r-Greedy(G, x, k) S := ∅ repeat k times Let u be a vertex which maximizes the value of πx/Zxx in G(V, E ∪ {(u, x)}) S := S ∪ {u} E := E ∪ {(u, x)} Report S as the solution

Lower bounds for the approximation ratios of greedy algorithms for LINK BUILDING In order to force a greater approximation ratio, we would have to consider graph families that use the independent set aspect of link building, as discussed in Remark1 and Section4. We want to construct a graph with vertices that have the following properties: k cycle vertices c1,c2,...,ck that – have high PageRank compared to their out-degree – form a cycle, and therefore are in the same community k sink vertices s1, s2,..., sk that – have PageRank values (compared to their degrees) slightly lower than the cycle vertices – do not belong to the same community – have links from the target vertex x towards them, and therefore are within a short distance from the target

Discussion and open problems We present a lower bound for the approximation ratio achieved by a perhaps more intuitive and simpler greedy algorithm. A more interesting open problem is to develop a polynomial time approximation scheme(PTAS) for LINK BUILDING