MINING DENSE SUBGRAPHS Andrew Lomonosov Presented by Thang
MINING DENSE SUBGRAPHS Andrew Lomonosov Presented by Thang N. Dinh
DENSE SUBGRAPH Dense = “More” edges than vertices.
DENSE SUBGRAPH (CONT. ) Arises in many CAD/CAM applications Geometric modeling contexts: Virtual reality Robotics Molecular modeling Teaching geometry Many applications require finding subgraphs which has twice the number of edges as vertices w(edge) = 1, w (vertices) = 2
STABLY DENSE GRAPH
EXTREME STABLY DENSE SUBGRAPH Dense Subgraph Problems 1. Minimal stably dense subgraph 2. Maximal stably dense subgraph 3. Minimum stably dense subgraph 4. Maximum stably dense subgraph Minimizing Edges Variation: Can be approximated via minimizing vertices Approximation ratio 3/2
MINIMIZING EDGES
EXAMPLES • K=0, w(edges) = 1, w(vertices) = 2 ü BCDE: minimal Stably Dense (SD) ü EDF: minimum SD ü AGH: maximal SD ü ABCDEF: maximum SD
REDUCTION BETWEEN DS PROBLEMS Minimum (minimal) DS Maximum (maximal) DS. Reverse reduction does not work
SUMMARY RESULTS
RELATED PROBLEMS
RELATED PROBLEMS • Feige: O(n/k) approximation algorithm via Semidefinite Programming for Finding subgraph < k vertices with maximum edges problem •
MAXIMUM DENSE SUBGRAPH LP: w(edges) = 1 w(vertices) = 3 LP gives Integral solution!
MINIMUM DENSE AS MINIMUM COST FLOW
- Slides: 14