Large-scale graph-structured data arising from social networks, databases, knowledge bases, web graphs, etc. is now available for analysis and mining. Graph-mining often involves 'relationship queries', which seek a ranked list of interesting interconnections among a given set of entities, corresponding to nodes in the graph. While relationship queries have been studied for many years, using various terminologies, e.g., keyword-search, Steiner-tree in a graph etc., the solutions proposed in the literature so far have not focused on scaling relationship queries to large graphs having billions of nodes and edges, such are now publicly available in the form of 'linked-open-data'. In this paper, we present an algorithm for distributed keyword search (DKS) on large graphs, based on the graph-parallel computing paradigm Pregel. We also present an analytical proof that our algorithm produces an optimally ranked list of answers if run to completion. Even if terminated early, our algorithm produces approximate answers along with bounds. We describe an optimized implementation of our DKS algorithm along with time-complexity analysis. Finally, we report and analyze experiments using an implementation of DKS on Giraph the graph-parallel computing framework based on Pregel, and demonstrate that we can efficiently process relationship queries on large-scale subsets of linked-open-data.
The Steiner Tree problem or Group Steiner Tree problem has been attempted in multiple domains such as for routing of network packets in computer networks, multiple applications in social networks @cite_13 @cite_28 , identification of functional modules in protein networks @cite_0 . Most of these algorithms are either a heuristic approach or apply a domain specific constraint to solve this problem.
@cite_28 @cite_0 @cite_13
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Given a weighted graph with some subsets of vertices called groups, the group Steiner tree problem is to find a minimum-weight subgraph which contains at least one vertex from each group. We give a randomized algorithm with a polylogarithmic approximation guarantee for the group Steiner tree problem. The previous best approximation guarantee was O(i2k1 i) in time O(nik2i) (Charikar, Chekuri, Goel, and Guha). Our algorithm also improves existing approximation results for network design problems with location-based constraints and for the symmetric generalized traveling salesman problem. The Steiner tree problem is one of the most fundamental NP-hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum-cost tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from 2 to the current best 1.55 [Robins,Zelikovsky-SIDMA'05]. All these algorithms are purely combinatorial. A long-standing open problem is whether there is an LP-relaxation for Steiner tree with integrality gap smaller than 2 [Vazirani,Rajagopalan-SODA'99]. In this paper we improve the approximation factor for Steiner tree, developing an LP-based approximation algorithm. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider a directed-component cut relaxation for the k-restricted Steiner tree problem. We sample one of these components with probability proportional to the value of the associated variable in the optimal fractional solution and contract it. We iterate this process for a proper number of times and finally output the sampled components together with a minimum-cost terminal spanning tree in the remaining graph. Our algorithm delivers a solution of cost at most ln(4) times the cost of an optimal k-restricted Steiner tree. This directly implies a ln(4)+e The Steiner tree problem is one of the most fundamental NP-hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum-cost tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from 2 to 1.55 [Robins and Zelikovsky 2005]. All these algorithms are purely combinatorial. A long-standing open problem is whether there is an LP relaxation of Steiner tree with integrality gap smaller than 2 [Rajagopalan and Vazirani 1999]. In this article we present an LP-based approximation algorithm for Steiner tree with an improved approximation factor. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider an LP relaxation of the problem, which is based on the notion of directed components. We sample one component with probability proportional to the value of the associated variable in a fractional solution: the sampled component is contracted and the LP is updated consequently. We iterate this process until all terminals are connected. Our algorithm delivers a solution of cost at most ln(4) + e As a by-product of our analysis, we show that the integrality gap of our LP is at most 1.55, hence answering the mentioned open question. A Steiner tree with maximum-weight edge minimized is called a bottleneck Steiner tree (BST). The authors propose a Theta ( mod rho mod log mod rho mod ) time algorithm for constructing a BST on a point set rho , with points labeled as Steiner or demand; a lower bound, in the linear decision tree model, is also established. It is shown that if it is desired to minimize further the number of used Steiner points, then the problem becomes NP-complete. It is shown that when locations of Steiner points are not fixed the problem remains NP-complete; however, if the topology of the final tree is given, then the problem can be solved in Theta ( mod rho mod log mod rho mod ) time. The BST problem can be used, for example, in VLSI layout, communication network design, and (facility) location problems. >