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Finding top-\(k\, r\)-cliques for keyword search from graphs in polynomial delay

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Abstract

Keyword search over structured data offers an alternative method to explore and query databases for users that are not familiar with the structure of the data and/or a query language. Structured data are usually modeled as graphs. In this context, an answer is a substructure of the graph that contains all or some of the query keywords. In most of the previous works, a minimal tree that covers all the query keywords are found as the answer. Some recent works offer to find subgraphs rather than minimal trees and show that subgraphs might be more informative for the users. However, current methods suffer from the following problems. Although some of the content nodes (i.e., nodes that contain input keywords) are close to each other in an answer, others might be far from each other. While searching for the best answer, current methods explore the whole graph rather than only the content nodes. This might increase the run time and leads to poor performance. To address these problems, we propose to find top-\(k\, r\)-cliques as the answers to the graph keyword search problem. An \(r\)-clique is a set of content nodes that cover all the input keywords, and the distance between each pair of nodes is less than or equal to \(r\). We propose a new weight function that is the sum of distances between each pair of content nodes. We prove that minimizing the new weight function is NP-hard and propose an approximation algorithm that produces \(r\)-cliques with 2-approximation ratio in polynomial delay. We further improve the run time of the approximation algorithm with the cost of increasing the approximation ratio. Extensive performance studies using three large real datasets confirm the efficiency and accuracy of finding \(r\)-cliques in graphs.

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Notes

  1. A content node is a node in the input graph that contains at least one of the input keywords.

  2. The shortest distance between two nodes in a graph is the sum of the weights of a path between the two nodes such that the sum of the weights of its constituent edges is minimized.

  3. The weight of the edges depend on the application’s dataset.

  4. A complete survey on keyword search in databases and graphs can be found in [25].

  5. For directed graphs, the shortest distance between two nodes in an \(r\)-clique should be no larger than \(r\) in both directions.

  6. 3-satisfiability (3-SAT) is the problem of determining whether there exists an interpretation that satisfies a given boolean formula in a conjunctive normal form where each clause is limited to at most three literals. See [24] for more details.

  7. There are other ways to prove this theorem, such as by reducing the problem to the multiple-choice cover problem. See [2] for the definition of the multiple-choice cover problem.

  8. It should be noted that the same approach is used in [2] for proving the NP-hardness of multiple-choice cover problem.

  9. The reason to choose this value will become clear later in the proof.

  10. Note that since the distance between \(u_i\) and \(\overline{u}_i\) is set to \(2\times w+\epsilon \) (where \(\epsilon >0\)) which is larger than \(r\) (set to \(2\times w\)), \(u_i\) and \(\overline{u}_i\) cannot be part of the same answer.

  11. Since the distance between other variables is set to \(\frac{w}{\small \big (\begin{array}{c}n+m\\ 2\end{array}\big )}\), and we want to choose \(n+m\) keywords, if a feasible solution exists, its weight is less than \(w\).

  12. Our approach to dividing a search space is similar to the idea used in [23].

  13. As we formally describe later, our approximation algorithms find an approximation of an \(r\)-clique. Therefore, in the rest of the paper, an approximation of an \(r\)-clique is called approx-\(r\)-clique.

  14. An inverted index is commonly used in information retrieval [3] and graph keyword search [23].

  15. This is also valid for approx-\(r\)-cliques.

  16. Note that the systems’s architecture is previously discussed in our demo paper published in [13]. It is briefly described here due to the completeness of this paper.

  17. In our implementation, we used quick sort.

  18. http://dblp.uni-trier.de/xml/.

  19. http://www.grouplens.org/node/73.

  20. http://www.dbis.informatik.uni-goettingen.de/Mondial/.

  21. We choose 8 users because it is mentioned in [21] that “For really low-overhead projects, it’s often optimal to test as little as 2 users per study. For some other projects, 8 users—or sometimes even more—might be better.”

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Authors and Affiliations

  1. Department of Computer Science and Engineering, York University, Toronto, Canada

    Mehdi Kargar & Aijun An

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  1. Mehdi Kargar
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  2. Aijun An
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Correspondence to Mehdi Kargar.

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Kargar, M., An, A. Finding top-\(k\, r\)-cliques for keyword search from graphs in polynomial delay. Knowl Inf Syst 43, 249–280 (2015). https://doi.org/10.1007/s10115-014-0736-0

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