A regret minimizing set Q is a small size representation of a much larger database P so that user queries executed on Q return answers whose scores are not much worse than those on the full dataset. In particular, a k-regret minimizing set has the property that the regret ratio between the score of the top-1 item in Q and the score of the top-k item in P is minimized, where the score of an item is the inner product of the item's attributes with a user's weight (preference) vector. The problem is challenging because we want to find a single representative set Q whose regret ratio is small with respect to all possible user weight vectors. We show that k-regret minimization is NP-Complete for all dimensions d >= 3. This settles an open problem from [VLDB 2014], and resolves the complexity status of the problem for all d: the problem is known to have polynomial-time solution for d <= 2. In addition, we propose two new approximation schemes for regret minimization, both with provable guarantees, one based on coresets and another based on hitting sets. We also carry out extensive experimental evaluation, and show that our schemes compute regret-minimizing sets comparable in size to the greedy algorithm proposed in [VLDB 14] but our schemes are significantly faster and scalable to large data sets.
The @math -RMS problem is also closely related to the problem of approximating the Pareto curve (or skyline) of a set of points. Papadamitriou and Yannakakis @cite_21 @cite_18 considered this problem and defined an approximate Pareto curve as a set of points whose @math scaling dominates every point on the skyline. They showed that there exists such a set of polynomial size @cite_21 @cite_18 . However, computing such a set of the smallest size is @math -Complete @cite_44 . See also @cite_30 .
@cite_44 @cite_18 @cite_21 @cite_30
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We propose and investigate from the algorithmic standpoint a novel form of fuzzy query called approximately dominating representatives or ADRs. The ADRs of a multidimensional point set consist of a few points guaranteed to contain an approximate optimum of any monotone Lipschitz continuous combining function of the dimensions. ADRs can be computed by appropriately post-processing Pareto, or ''skyline'', queries [Kian-Lee Tan, Pin-Kwang Eng, Beng Chin Ooi, Efficient progressive skyline computation, in: VLDB, 2001, pp. 301-310; Wolf-Tilo Balke, Ulrich Guntzer, Jason Xin Zheng, Efficient distributed skylining for web information systems, in: EDBT, 2004. [14]]. We show that the problem of minimizing the number of points returned, for a user-specified desired approximation, can be solved in polynomial time in two dimensions; for three and more it is NP-hard but has a polynomial-time logarithmic approximation. Finally, we present a polynomial-time, constant factor approximation algorithm for three dimensions. The optimization of queries in distributed database systems is known to be subject to delicate trade-offs. For example, the Mariposa database system allows users to specify a desired delay-cost tradeoff (that is, to supply a decreasing function u(d), specifying how much the user is willing to pay in order to receive the query results within time d); Mariposa divides a query graph into horizontal “strides,” analyzes each stride, and uses a greedy heuristic to find the “best” plan for all strides. We show that Mariposa's greedy heuristic can be arbitrarily far from the desired optimum. Applying a recent approach in multiobjective optimization algorithms to this problem, we show that the optimum cost-delay trade-off (Pareto) curve in Mariposa's framework can be approximated fast within any desired accuracy. We also present a polynomial algorithm for the general multiobjective query optimization problem, which approximates arbirarily well the optimum cost-delay tradeoff (without the restriction of Mariposa's heuristic stride subdivision). We study problems in multiobjective optimization, in which solutions to a combinatorial optimization problem are evaluated with respect to several cost criteria, and we are interested in the trade-off between these objectives (the so-called Pareto curve). We point out that, under very general conditions, there is a polynomially succinct curve that spl epsiv -approximates the Pareto curve, for any spl epsiv >0. We give a necessary and sufficient condition under which this approximate Pareto curve can be constructed in time polynomial in the size of the instance and 1 spl epsiv . In the case of multiple linear objectives, we distinguish between two cases: when the underlying feasible region is convex, then we show that approximating the multi-objective problem is equivalent to approximating the single-objective problem. If however the feasible region is discrete, then we point out that the question reduces to an old and recurrent one: how does the complexity of a combinatorial optimization problem change when its feasible region is intersected with a hyperplane with small coefficients; we report some interesting new findings in this domain. Finally, we apply these concepts and techniques to formulate and solve approximately a cost-time-quality trade-off for optimizing access to the World-Wide Web, in a model first studied by (1996) (which was actually the original motivation for this work). Trade-off (aka Pareto) curves are typically used to represent the trade-off among different objectives in multiobjective optimization problems. Although trade-off curves are exponentially large for typical combinatorial optimization problems (and infinite for continuous problems), it was observed in Papadimitriou and Yannakakis [On the approximability of trade-offs and optimal access of web sources, in: Proc. 41st IEEE Symp. on Foundations of Computer Science, 2000] that there exist polynomial size e approximations for any e > 0, and that under certain general conditions, such approximate e-Pareto curves can be constructed in polynomial time. In this paper we seek general-purpose algorithms for the efficient approximation of trade-off curves using as few points as possible. In the case of two objectives, we present a general algorithm that efficiently computes an e-Pareto curve that uses at most 3 times the number of points of the smallest such curve; we show that no algorithm can be better than 3-competitive in this setting. If we relax e to any e' > e, then we can efficiently construct an e'-curve that uses no more points than the smallest e-curve. With three objectives we show that no algorithm can be c-competitive for any constant c unless it is allowed to use a larger e value. We present an algorithm that is 4-competitive for any e' > (1 + e)2 - 1. We explore the problem in high dimensions and give hardness proofs showing that (unless P=NP) no constant approximation factor can be achieved efficiently even if we relax e by an arbitrary constant.