M INING D ENSE S UBGRAPH S 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 S TABLY D ENSE G RAPH E XTREME S TABLY D ENSE S UBGRAPH 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 M INIMIZING E DGES E XAMPLES BCDE: minimal Stably Dense (SD) EDF: minimum SD AGH: maximal SD ABCDEF: maximum SD K=0, w(edges) = 1, w(vertices) = 2 R EDUCTION B ETWEEN DS P ROBLEMS Minimum (minimal) DS Maximum (maximal) DS. Reverse reduction does not work S UMMARY R ESULTS R ELATED P ROBLEMS Feige: O(n/k) approximation algorithm via Semidefinite Programming for Finding subgraph < k vertices with maximum edges problem M AXIMUM D ENSE S UBGRAPH LP: w(edges) = 1 w(vertices) = 3 LP gives Integral solution! M INIMUM D ENSE AS M INIMUM C OST F LOW