Large-scale Multi-dimensional Assignment: Problem Formulations and GPU Accelerated Solutions

In this paper, we present alternate integer programming formulations for the multi-dimensional assignment problem, which is typically employed for multi-sensor fusion, multi-target tracking (MTT) or data association in general. The first formulation is the Axial Multidimensional Assignment Problem with Decomposable Costs (MDADC). The decomposable costs comes from the fact that there are only pairwise costs between stages or scans of a target tracking problem or corpuses of a data association context. The difficulty with this formulation is the large number of transitivity or triangularity constraints that ensure if entity $A$ is associated to entity $B$ and entity $B$ is associated with entity $C$, then it must also be that entity $A$ is associated to entity $C$. The second formulation uses both pairs and triplets of observations, which offer more accurate representation for kinematic tracking of targets. This formulation avoids the large number of transitivity constraints but significantly increases the number of variables due to triples. Solution to large-scale problems has alluded researchers and practitioners alike. We present solution methods based on Lagrangian Relaxation and massively parallel algorithms that are implemented on Graphics Processing Units (GPUs). We test the problem formulations and solution algorithms on MTT problems. The triples formulation tends to be more accurate for tracking measures and the MDADC solver can solve much larger problems in reasonable computational time.