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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 -Happiness Maximizing Sets (-HMS) problems.
Specifically, AHMS maximizes the average of this ratio within a distribution of
utility functions. Meanwhile, -HMS selects records from the database
such that the minimum happiness ratio between the -th best score in the
database and the best score in the selected records for any possible utility
function is maximized. AHMS and -HMS seek the same optimal solutions as the
more established Average Regret Minimizing Sets (ARMS) and -Regret
Minimizing Sets (-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 -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 -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
-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 -RMS solvers by up to 92\%