A differentiable approximation for the Linear Sum Assignment Problem with Edition.

Linear Sum Assignment Problem (LSAP) consists in mapping two sets of points of equal sizes according to a matrix encoding the cost of mapping each pair of points. The Linear Sum Assignment Problem with Edition (LSAPE) extends this problem by allowing the mapping of sets of different sizes and adding the possibility to reject some matchings. This problem is set up by a rectangular cost matrix whose last column and last line encode the costs of rejecting the match of an element of respectively the first and the second sets. LSAPE has been the workhorse of many fundamental graph problems such as graph edit distance, median graph computation or sub graph matching. LSAP may be solved using the Hungarian algorithm while an equivalent efficient discrete algorithm has been designed for LSAPE. However, while the Sinkhorn algorithm constitutes a continuous solver for LSAP, no such algorithm yet exists for LSAPE. This lack of solvers forbids the integration of LSAPE in Neural networks requiring continuous operations from the input to the final loss. This paper aims at providing such a solver, hence paving the way to an integration of LSAPE solvers in Neural Networks.