# Finding dense subgraphs with semidefinite programming

## Abstract

In this paper we consider the problem of computing the heaviest *k*-vertex induced subgraph of a given graph with nonnegative edge weights. This problem is known to be *NP*-hard, but its approximation complexity is not known. For the general problem only an approximation ratio of Õ(n^{0.3885}) has been proved (Kortsarz and Peleg (1993)). In the last years several authors analyzed the case *k=Ω(n)*. In this case Asahiro et al. (1996) showed a constant factor approximation, and for dense graphs Arora et al. (1995) obtained even a polynomial-time approximation scheme. We give a new approximation algorithm for arbitrary graphs and *k=n/c* for *c* > 1 based on semidefinite programming and randomized rounding which achieves for some c the presently best (randomized) approximation factors.

## Key words

Subgraph Problem Approximation Algorithms Randomized Algorithms Semidefinite Programming## Preview

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