Selecting a subset of the $k$ "best" items from a dataset of $n$ items, based on a scoring function, is a key task in decision-making. Given the rise of automated decision-making software, it is important that the outcome of this process, called top-$k$ selection, is fair. Here we consider the problem of identifying a fair linear scoring function for top-$k$ selection. The function computes a score for each item as a weighted sum of its (numerical) attribute values, and must ensure that the selected subset includes adequate representation of a minority or historically disadvantaged group. Existing algorithms do not scale efficiently, particularly in higher dimensions. Our hardness analysis shows that in more than two dimensions, no algorithm is likely to achieve good scalability with respect to dataset size, and the computational complexity is likely to increase rapidly with dimensionality. However, the hardness results also provide key insights guiding algorithm design, leading to our two-pronged solution: (1) For small values of $k$, our hardness analysis reveals a gap in the hardness barrier. By addressing various engineering challenges, including achieving efficient parallelism, we turn this potential of efficiency into an optimized algorithm delivering substantial practical performance gains. (2) For large values of $k$, where the hardness is robust, we employ a practically efficient algorithm which, despite being theoretically worse, achieves superior real-world performance. Experimental evaluations on real-world datasets then explore scenarios where worst-case behavior does not manifest, identifying areas critical to practical performance. Our solution achieves speed-ups of up to several orders of magnitude compared to SOTA, an efficiency made possible through a tight integration of hardness analysis, algorithm design, practical engineering, and empirical evaluation.

The goal in the area of functions property testing is to determine whether a given black-box Boolean function has a particular given property or is ɛ-far from having that property. We investigate here several types of properties testing for Boolean functions (identity, correlations and balancedness) using the Deutsch-Jozsa algorithm (for the Deutsch-Jozsa (D-J) problem) and also the amplitude amplification technique. At first, we study here a particular testing problem: namely whether a given Boolean function f, of n variables, is identical with a given function g or is ɛ-far from g, where ɛ is the parameter. We present a one-sided error quantum algorithm to deal with this problem that has the query complexity [Formula: see text]. Moreover, we show that our quantum algorithm is optimal. Afterwards we show that the classical randomized query complexity of this problem is [Formula: see text]. Secondly, we consider the D-J problem from the perspective of functional correlations and let C( f, g) denote the correlation of f and g. We propose an exact quantum algorithm for making distinction between | C( f, g)| = ɛ and | C( f, g)| = 1 using six queries, while the classical deterministic query complexity for this problem is Θ(2 n ) queries. Finally, we propose a one-sided error quantum query algorithm for testing whether one Boolean function is balanced versus ɛ-far balanced using [Formula: see text] queries. We also prove here that our quantum algorithm for balancedness testing is optimal. At the same time, for this balancedness testing problem we present a classical randomized algorithm with query complexity of O(1/ ɛ 2 ). Also this randomized algorithm is optimal. Besides, we link the problems considered here together and generalize them to the general case.