In recent years, sparsity and compressive sensing have attracted significant attention in parameter estimation tasks, including frequency estimation, delay estimation, and localization. Parametric dictionaries collect signals for a sampling of the parameter space and can yield sparse representations for the signals of interest when the sampling is sufficiently dense. While this dense sampling can lead to high coherence in the dictionary, it is possible to leverage structured sparsity models to prevent highly coherent dictionary elements from appearing simultaneously in a signal representation, alleviating these coherence issues. However, the resulting approaches depend heavily on a careful setting of the maximum allowable coherence; furthermore, their guarantees apply to the coefficient vector recovery and do not translate in general to the parameter estimation task. We propose a new algorithm based on optimal sparse approximation measured by earth mover's distance (EMD). Theoretically, we show that EMD provides a better metric for the performance of parametric dictionary-based parameter estimation and $K$-median clustering algorithms has the potential to solve the EMD-optimal sparse approximation problems. Simulations show that the resulting compressive parameter estimation algorithm is better at addressing the coherence issuers without a careful setting of additional parameters.

In recent years there has been a trend towards large-scale logistics for individual members of the public, such as ride-sharing services and drone package delivery. Efficient coordination of pickups and deliveries is essential in order to keep costs and wait times down. In this thesis we present these types of problems in a more general framework, expanding applicability of our discussion to an even wider domain of problems. We present fast new al- gorithms with supporting theoretical and experimental analysis, providing certain guarantees about how close our algorithms compare to a theoretically optimal approach. / Master of Science