A Regret Minimizing Set (RMS) is a useful concept in which a smaller subset of a database is selected while mostly preserving the best scores along every possible utility function. In this paper, we study the $k$-Regret Minimizing Sets ($k$-RMS) and Average Regret Minimizing Sets (ARMS) problems. $k$-RMS selects $r$ records from a database such that the maximum regret ratio between the $k$-th best score in the database and the best score in the selected records for any possible utility function is minimized. Meanwhile, ARMS minimizes the average of this ratio within a distribution of utility functions. Particularly, we study approximation algorithms for $k$-RMS and ARMS from the perspective of approximating the happiness ratio, which is equivalent to one minus the regret ratio. In this paper, we show that the problem of approximating the happiness of a $k$-RMS within any finite factor is NP-Hard when the dimensionality of the database is unconstrained and extend the result to an inapproximability proof for the regret. We then provide approximation algorithms for approximating the happiness of ARMS with better approximation ratios and time complexities than known algorithms for approximating the regret. We further provide dataset reduction schemes which can be used to reduce the runtime of existing heuristic based algorithms, as well as to derive polynomial-time approximation schemes for $k$-RMS when dimensionality is fixed. Finally, we provide experimental validation.