A Sampling Strategy for Projecting to Permutations in the Graph Matching Problem

In the context of the graph matching problem we propose a novel method for projecting a matrix $Q$, which may be a doubly stochastic matrix, to a permutation matrix $P.$ We observe that there is an intuitve mapping, depending on a given $Q,$ from the set of $n$-dimensional permutation matrices to sets of points in $\mathbb{R}^n$. The mapping has a number of geometrical properties that allow us to succesively sample points in $\mathbb{R}^n$ in a manner similar to simulated annealing, where our objective is to minimise the graph matching norm found using the permutation matrix corresponding to each of the points. Our sampling strategy is applied to the QAPLIB benchmark library and outperforms the PATH algorithm in two-thirds of cases. Instead of using linear assignment, the incorporation of our sampling strategy as a projection step into algorithms such as PATH itself has the potential to achieve even better results.