Finding Dense Subgraphs with Size Bounds

  • Authors:

  • Reid Andersen;Kumar Chellapilla

  • Affiliations:

  • Microsoft Live Labs, Redmond, USA WA 98052;Microsoft Live Labs, Redmond, USA WA 98052

  • Venue:
  • WAW '09 Proceedings of the 6th International Workshop on Algorithms and Models for the Web-Graph
  • Year:
  • 2009

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Abstract

We consider the problem of finding dense subgraphs with specified upper or lower bounds on the number of vertices. We introduce two optimization problems: the densest at-least-k-subgraph problem (dalks), which is to find an induced subgraph of highest average degree among all subgraphs with at least k vertices, and the densest at-most-k-subgraph problem (damks), which is defined similarly. These problems are relaxed versions of the well-known densest k-subgraph problem (dks), which is to find the densest subgraph with exactly k vertices. Our main result is that dalks can be approximated efficiently, even for web-scale graphs. We give a (1/3)-approximation algorithm for dalks that is based on the core decomposition of a graph, and that runs in time O(m + n), where n is the number of nodes and m is the number of edges. In contrast, we show that damks is nearly as hard to approximate as the densest k-subgraph problem, for which no good approximation algorithm is known. In particular, we show that if there exists a polynomial time approximation algorithm for damks with approximation ratio γ, then there is a polynomial time approximation algorithm for dks with approximation ratio γ 2/8. In the experimental section, we test the algorithm for dalks on large publicly available web graphs. We observe that, in addition to producing near-optimal solutions for dalks, the algorithm also produces near-optimal solutions for dks for nearly all values of k.