### Abstract

The matching problem between two adjacency matrices can be formulated as the NP-hard quadratic assignment problem (QAP). Previous work on semidefinite programming (SDP) relaxations to the QAP have produced solutions that are often tight in practice, but such SDPs typically scale badly, involving matrix variables of dimension n^{2} where n is the number of nodes. To achieve a speed up, we propose a further relaxation of the SDP involving a number of positive semidefinite matrices of dimension O(n) no greater than the number of edges in one of the graphs. The relaxation can be further strengthened by considering cliques in the graph, instead of edges. The dual problem of this novel relaxation has a natural three-block structure that can be solved via a convergent Alternating Direction Method of Multipliers in a distributed manner, where the most expensive step per iteration is computing the eigendecomposition of matrices of dimension O(n). The new SDP relaxation produces strong bounds on quadratic assignment problems where one of the graphs is sparse with reduced computational complexity and running times, and can be used in the context of nuclear magnetic resonance spectroscopy to tackle the assignment problem.

Original language | English (US) |
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Pages (from-to) | 677-712 |

Number of pages | 36 |

Journal | Computational Optimization and Applications |

Volume | 69 |

Issue number | 3 |

DOIs | |

State | Published - Apr 1 2018 |

### All Science Journal Classification (ASJC) codes

- Control and Optimization
- Computational Mathematics
- Applied Mathematics

### Keywords

- Alternating direction method of multipliers
- Convex relaxation
- Graph matching
- Quadratic assignment problem
- Semidefinite programming

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## Cite this

*Computational Optimization and Applications*,

*69*(3), 677-712. https://doi.org/10.1007/s10589-017-9968-8