## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Thursday, December 01, 2005, 12:15 pm

Duration: This information is not available in the database

Location: This information is not available in the database

Speaker: Takeaki Uno (National Institute of Informatics (NII)

## Enumerating Dense Subgraphs

For a graph G=(V,E) and its vertex set S, the density of the induced subgraph by S is |E(G[S])| / ((|S|-1)|S|/2) where |E(G[S])| is the number of edges in the induced subgraph, and ((|S|-1)|S|/2) is the maximum number of edges can be included in the induced subgraph of |S| vertices (edges in the clique of size |S|). S is an independent set if its density is 0 and is a clique if its density is 1. For a given constant e, 0 \le e \le 1, S is called dense subgraph if its density is no less than e.

In this talk, we consider the problem of enumerating all dense subgraphs from a graph with threshold e. This problem comes from data mining, data engineering, AI, etc. In these areas, modeling by cliques is a popular way, so they want to enumerate cliques possibly with constraints. However, usually the real data includes some error, so the vertex sets should be cliques often lose some edges in the data. So, finding "clique like structure" is important.

For this enumeration problem, we show 1. straightforward approach is hard, 2. polynomial time algorithm based on reverse search, 3. result of computational experiments.

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