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 n2 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 Augmented Direction Method of Multipliers (ADMM) 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 (NMR)
to tackle the assignment problem.Comment: 31 page