Matching hypergraphs with game theory

Abstract

The problem of finding isomorphisms, or matching partitions, in hypergraphs has gained increasing interest from the scientific community in the last years, particularly in the Computer Vision field. This is due to the advantages that arises from overcoming the limitations provided by pairwise relationship, thus encoding a bigger pool of information. Association graph techniques represent a classical approach to tackle the graph matching problem and recently the idea has been generalized to the case of uniform hypergraphs. In this thesis, the potential of this approach, employed together with elements from the Evolutionary Game Theory, is explored. Indeed, the proposed framework uses a class of dynamical systems derived from the Baum-Eagon inequality in order to find the maximum (maximal) clique in the association hypergraph, that corresponds to the maximum (maximal) isomorphism between the hypergraphs to be matched. The proposed approach has extensively been tested with experiments on a large synthetic dataset. In particular both the pure isomorphism and the subgraph isomorphism problems have been analysed. The obtained results reflect the different complexity classes these problems belong to, thus showing that despite its simplicity the Baum-Eagon dynamics does an excellent job at finding globally optimal solutions in the pure isomorphism case, while in the subgraph case the use of more complex dynamics might be more suitable.