Approximating Happiness Maximizing Set Problems

A Happiness Maximizing Set (HMS) 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 Average Happiness Maximizing Sets (AHMS) and $k$-Happiness Maximizing Sets ($k$-HMS) problems. Specifically, AHMS maximizes the average of this ratio within a distribution of utility functions. Meanwhile, $k$-HMS selects $r$ records from the database such that the minimum happiness ratio between the $k$-th best score in the database and the best score in the selected records for any possible utility function is maximized. AHMS and $k$-HMS seek the same optimal solutions as the more established Average Regret Minimizing Sets (ARMS) and $k$-Regret Minimizing Sets ($k$-RMS) problems, respectively, but the use of the happiness metric allows for the derivation of stronger theoretical results and more natural approximation schemes. We provide approximation algorithms for AHMS with better approximation ratios and time complexities than known algorithms for ARMS. Next, we show that the problem of approximating $k$-HMS within any finite factor is NP-Hard when the dimensionality of the database is unconstrained and extend the result to an inapproximability proof of $k$-RMS. Finally, we 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 both $k$-HMS when dimensionality is fixed. We further provide experimental validation showing that our AHMS algorithm achieves the same happiness as the existing Greedy Shrink FAM algorithm while running faster by over 2 orders of magnitude on even small datasets while our reduction scheme was able to reduce runtimes of existing $k$-RMS solvers by up to 92\%.