Graph Transformation Based Guidance for Web Navigation

With growing information volume and diverse user preferences on the web, the performance of web information retrieval has become a critical issue. Web navigation is dramatically influenced by the organizations of web contents. Hence, useful navigation guidance can considerably accelerate the information retrieval process. In this paper, web navigation is formulated as a Directed Group Steiner Forest (DGSF) problem in line graph representation of the website. A heuristic algorithm is proposed to tackle the DGSF problem and attain the suboptimal solution in polynomial time. Simulations are conducted to compare the mean searching time for the proposed DGSF-based navigation guidance and other approaches. The results suggest that the DGSF-based navigation guidance can significantly reduce the mean searching time, especially when the number of web pages is large while the number of destination pages is moderate. The discussion is also made for extending the model to take into account the websites owner’s interests and other concerns as well

Approximation Algorithms for (S,T)-Connectivity Problems

We study a directed network design problem called the $k$-$(S,T)$-connectivity problem; we design and analyze approximation algorithms and give hardness results. For each positive integer $k$, the minimum cost $k$-vertex connected spanning subgraph problem is a special case of the $k$-$(S,T)$-connectivity problem. We defer precise statements of the problem and of our results to the introduction. For $k=1$, we call the problem the $(S,T)$-connectivity problem. We study three variants of the problem: the standard $(S,T)$-connectivity problem, the relaxed $(S,T)$-connectivity problem, and the unrestricted $(S,T)$-connectivity problem. We give hardness results for these three variants. We design a $2$-approximation algorithm for the standard $(S,T)$-connectivity problem. We design tight approximation algorithms for the relaxed $(S,T)$-connectivity problem and one of its special cases. For any $k$, we give an $O(\log k\log n)$-approximation algorithm, where $n$ denotes the number of vertices. The approximation guarantee almost matches the best approximation guarantee known for the minimum cost $k$-vertex connected spanning subgraph problem which is $O(\log k\log\frac{n}{n-k})$ due to Nutov in 2009

Improved approximating algorithms for directed steiner forest

We consider the k-Directed Steiner Forest (k-DSF) problem: Given a directed graph G = (V, E) with edge costs, a collection D ⊆ V × V of ordered node pairs, and an integer k ≤ |D|, find a minimum cost subgraph H of G that contains an st-path for (at least) k pairs (s, t) ∈ D. When k = |D|, we get the Directed Steiner Forest (DSF) problem. The best known approximation ratios for these problems are: Õ(k2/3) for k-DSF by Charikar et al. [6], and O(k 1/2+ε) for DSF by Chekuri et al. [7]. Our main result is achieving the first sub-linear in terms of n = |V | approximation ratio for DSF. Specifically, we give an O(n ε · min{n 4/5, m 2/3})-approximation scheme for DSF. For k-DSF we give a simple greedy O(k 1/2+ε)-approximation algorithm. This improves upon the best known ratio Õ(k2/3) by Charikar et al. [6], and (almost) matches, in terms of k, the best ratio known for the undirected variant [18]. This algorithm uses a new structure called start-junction tree which may be of independent interest

Improved approximating algorithms for directed steiner forest

We consider the k-Directed Steiner Forest (k-DSF) problem: Given a directed graph G = (V, E) with edge costs, a collection D ⊆ V × V of ordered node pairs, and an integer k ≤ |D|, find a minimum cost subgraph H of G that contains an st-path for (at least) k pairs (s, t) ∈ D. When k = |D|, we get the Directed Steiner Forest (DSF) problem. The best known approximation ratios for these problems are: Õ(k2/3) for k-DSF by Charikar et al. [6], and O(k 1/2+ε) for DSF by Chekuri et al. [7]. Our main result is achieving the first sub-linear in terms of n = |V | approximation ratio for DSF. Specifically, we give an O(n ε · min{n 4/5, m 2/3})-approximation scheme for DSF. For k-DSF we give a simple greedy O(k 1/2+ε)-approximation algorithm. This improves upon the best known ratio Õ(k2/3) by Charikar et al. [6], and (almost) matches, in terms of k, the best ratio known for the undirected variant [18]. This algorithm uses a new structure called start-junction tree which may be of independent interest