We propose a method that automatically tracks and segments living cells in phase-contrast image sequences, especially for cells that deform and interact with each other or clutter. We formulate the problem as a many-to-one elastic partial matching problem between closed curves. We introduce Double Cyclic Dynamic Time Warping for the scenario where a collision event yields a single boundary that encloses multiple touching cells and that needs to be cut into separate cell boundaries. The resulting individual boundaries may consist of segments to be connected to produce closed curves that match well with the individual cell boundaries before the collision event. We show how to convert this partial-curve matching problem into a shortest path problem that we then solve efficiently by reusing the computed shortest path tree. We also use our shortest path algorithm to fill the gaps between the segments of the target curves. Quantitative results demonstrate the benefit of our method by showing maintained accurate recognition of individual cell boundaries across 8068 images containing multiple cell interactions.