近似后悔最小化集：一个幸福的视角
Approximating Regret Minimizing Sets: A Happiness Perspective
摘要：
遗憾最小化集（RMS）是一个有用的概念，其中选择数据库的较小子集，同时沿着每个可能的效用函数保留最佳分数。本文研究了$k$ -后悔最小集（$k$ -RMS）和平均后悔最小集（ARMS）问题. $k$ -RMS从数据库中选择$r$记录，使得对于任何可能的效用函数，数据库中第$k$个最佳分数与所选记录中的最佳分数之间的最大遗憾比率最小化。同时，ARMS在效用函数的分布内最小化该比率的平均值。特别是，我们研究近似算法的$k$ -RMS和ARMS的角度来看，近似的幸福比，这是相当于一减去遗憾比。在本文中，我们证明了问题的近似幸福的$k$ -RMS在任何有限的因素是NP-困难的数据库的维数是无约束的，并扩展结果的不可近似性证明遗憾。然后，我们提供了近似算法近似的幸福ARMS更好的近似比和时间复杂度比已知的算法近似的遗憾。我们还提供了数据集减少计划，可用于减少现有的启发式算法的运行时间，以及推导多项式时间近似方案时，维数是固定的$k$ -RMS。最后，我们提供了实验验证。
英文摘要：
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.
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