### Abstract

The maximum clique problem is a classical problem in combinatorial optimization that has a broad range of applications in graph-based data mining, social and biological network analysis and a variety of other fields. This article investigates the problem when the edges fail independently with known probabilities. This leads to the maximum probabilistic clique problem, which is to find a subset of vertices of maximum cardinality that forms a clique with probability at least θ [ 0, 1 ], which is a user-specified probability threshold. We show that the probabilistic clique property is hereditary and extend a well-known exact combinatorial algorithm for the maximum clique problem to a sampling-free exact algorithm for the maximum probabilistic clique problem. The performance of the algorithm is benchmarked on a test-bed of DIMACS clique instances and on a randomly generated test-bed.

Original language | English |
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Pages (from-to) | 105-120 |

Number of pages | 16 |

Journal | Journal of Combinatorial Optimization |

Volume | 28 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2014 |

Externally published | Yes |

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### Keywords

- Branch-and-bound
- Maximum clique problem
- Probabilistic clique
- Probabilistic programming

### Cite this

*Journal of Combinatorial Optimization*,

*28*(1), 105-120. https://doi.org/10.1007/s10878-013-9699-4