  Applications, ranging from tracking molecular motion within cells to
analyzing complex animal foraging behavior, require algorithms for associating
a collection of spot-like particles in one image with particles contained in
another image. These associations are often made via network flow algorithms.
However, it is often the case that many candidate association solutions (the
output of network flow algorithms) have nearly optimal scores; in this case,
the optimal assignment solution is of dubious quality. Algorithms for reliably
computing the uncertainty of candidate association solutions are
under-developed in situations where many particles are tracked over multiple
frames of data. This is due in part to the fact that exact uncertainty
quantification (UQ) in large association problems is computationally
intractable because the exact computation exhibits exponential dependence on
the number of particles tracked. We introduce a technique that can accurately
and efficiently quantify association ambiguity (i.e., UQ for discrete
association problems) without requiring the evaluation of the cost of each
feasible association solution. Our method can readily be wrapped around
existing tracking algorithms and can efficiently handle a variety of 2D
association problems. The applications presented are focused on tracking
molecules in live cells. Our method is validated via both simulations and
experiments. The experimental applications aim to accurately form tracks and
quantify diffusion of quantum dot labeled proteins from \emph{in vivo}
measurements; here association problems involving cost matrices possessing
hundreds to thousands of rows/columns are encountered. For such large-scale
problems, we discuss how our approach can efficiently and accurately quantify
inherent uncertainty in candidate data associations.
