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**1 - 1**of**1**### A Novel Approach to Finding Near-Cliques: The Triangle-Densest Subgraph Problem

, 2014

"... Many graph mining applications rely on detecting subgraphs which are large near-cliques. There exists a dichotomy between the results in the existing work related to this problem: on the one hand formulations that are geared towards finding large near-cliques are NP-hard and frequently inapproximabl ..."

Abstract
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Many graph mining applications rely on detecting subgraphs which are large near-cliques. There exists a dichotomy between the results in the existing work related to this problem: on the one hand formulations that are geared towards finding large near-cliques are NP-hard and frequently inapproximable due to connections with the Maximum Clique problem. On the other hand, the densest subgraph problem (DS-Problem) which maximizes the average degree over all subgraphs and other indirect approaches which optimize tractable objectives fail to detect large near-cliques in many networks. In this work, we propose a formulation which combines the best of both worlds: it is solvable in polynomial time and succeeds consistently in finding large near-cliques. Surprisingly, our formulation is a simple variation of the DS-Problem. Specifically, we define the triangle densest subgraph problem (TDS-Problem): given a graph G(V,E), find a subset of vertices S ∗ such that τ(S∗) = max S⊆V t(S) |S | , where t(S) is the number of triangles induced by the set S. We provide various exact and approximation algorithms which the solve TDS-Problem efficiently. Furthermore, we show how our algorithms adapt to the more general problem of maximizing the k-clique average density, k ≥ 2. We illustrate the success of the proposed formulation in extracting large near-cliques from graphs by performing numerous experiments on real-world networks. 1