Faster Algorithms for k -Regret Minimizing Sets via Monotonicity and Sampling

Efficient processing of k-regret minimization queries with theoretical guarantees

“…These bounds or theoretical results focus on the regret ratio instead of approximation guarantees. Qiu et al [8] and Dong et al [9] also provided the sampling techniques for 1-RMS and k-RMS queries, respectively. However, the proposed method in [8] is a special case of ours, where the sample factor equals 1, i.e., λ = 1.…”

“…Moreover, the concept of the regret ratio is also adopted to solve the problems in the machine learning area, e.g., multi-objective submodular function maximization [40]. Due to the hardness of the problem, most of the studies utilize the greedy framework [4], [5], [6], [7], [8], [9]. However, there is still no strict theoretical analysis of the approximation ratio in the original greedy framework.…”

“…Monotone nondecreasing submodular functions have been shown with an 1 − 1/e approximation ratio and can be well approximated, i.e., near optimal [17]. Because H D (S, u) is submodular while H D (S, U) is not, we introduce the submodularity ratio and generalized curvature to model H D (S, U) to obtain the guarantees of the greedy framework for regret minimization (or happiness maximization) queries [4], [5], [6], [7], [8], [9]. Lemma 1.…”

“…Due to the NP-hardness of the problem, in [3], RDPGREED was proposed based on the Ramer-Douglas-Peucker greedy framework. The greedy strategy has also been used in most follow-up studies, e.g., [4], [5], [6], [7], [8], [9].…”

Happiness Maximizing Sets under Group Fairness Constraints (Technical Report)

“…There have been many studies on regret minimizing set (RMS) [5,35,36,41,55], happiness maximizing set (HMS) [39,56], and different variants of them [2,6,8,12,14,15,29,30,34,38,43,44,50,52,53,59] (see [54] for an extensive survey). The RMS problem was first proposed by Nanongkai et al [35] to alleviate the deficiencies of top-𝑘 and skyline queries.…”