GRMR：广义后悔最小化代表
GRMR: Generalized Regret-Minimizing Representatives
摘要：
从大型数据库中抽取代表元组的子集是多准则决策中的一项重要任务。遗憾最小化集（RMS）问题是最近提出的代表性发现从数据库。具体地，对于$d$维度中的元组（点）的集合，RMS问题找到最小子集，使得对于任何可能的排名函数，子集中排名最高的点与整个数据库中排名最高的点之间的分数的相对差在参数$\varepsilon \in (0,1)$内。虽然RMS和它的变化已经在文献中进行了广泛的研究，现有的方法只考虑类的非负（单调）的线性函数的排名，这在建模用户的偏好和决策过程中有局限性。为了解决这个问题，我们定义了广义遗憾最小化代表（GRMR）的问题，考虑到所有的线性函数，包括非单调的负权重扩展RMS。对于二维数据库，我们提出了一个最优算法GRMR通过转换成最短的周期问题的有向图。由于GRMR被证明是NP-困难的，即使在三维，我们进一步开发了一个多项式时间的启发式算法的GRMR数据库中的任意尺寸。最后，我们在真实和合成数据集上进行了广泛的实验，以确认我们提出的算法的效率，有效性和可扩展性。
英文摘要：
Extracting a small subset of representative tuples from a large database is an important task in multi-criteria decision making. The regret-minimizing set (RMS) problem is recently proposed for representative discovery from databases. Specifically, for a set of tuples (points) in $d$ dimensions, an RMS problem finds the smallest subset such that, for any possible ranking function, the relative difference in scores between the top-ranked point in the subset and the top-ranked point in the entire database is within a parameter $\varepsilon \in (0,1)$. Although RMS and its variations have been extensively investigated in the literature, existing approaches only consider the class of nonnegative (monotonic) linear functions for ranking, which have limitations in modeling user preferences and decision-making processes. To address this issue, we define the generalized regret-minimizing representative (GRMR) problem that extends RMS by taking into account all linear functions including non-monotonic ones with negative weights. For two-dimensional databases, we propose an optimal algorithm for GRMR via a transformation into the shortest cycle problem in a directed graph. Since GRMR is proven to be NP-hard even in three dimensions, we further develop a polynomial-time heuristic algorithm for GRMR on databases in arbitrary dimensions. Finally, we conduct extensive experiments on real and synthetic datasets to confirm the efficiency, effectiveness, and scalability of our proposed algorithms.
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