You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
We consider stochastic settings for clustering, and develop provably-good (approximation) algorithms for a number of these notions. These algorithms allow one to obtain better approximation ratios compared to the usual deterministic clustering setting. Additionally, they offer a number of advantages including providing fairer clustering and clustering which has better long-term behavior for each user. In particular, they ensure that *every user* is guaranteed to get good service (on average). We also complement some of these with impossibility results.

Cited Abstract(s):
[1]: What does it mean for an algorithm to be biased? In U.S. law, unintentional bias is encoded via disparate impact, which occurs when a selection process has widely different outcomes for different groups, even as it appears to be neutral. This legal determination hinges on a definition of a protected class (ethnicity, gender) and an explicit description of the process. When computers are involved, determining disparate impact (and hence bias) is harder. It might not be possible to disclose the process. In addition, even if the process is open, it might be hard to elucidate in a legal setting how the algorithm makes its decisions. Instead of requiring access to the process, we propose making inferences based on the data it uses. We present four contributions. First, we link disparate impact to a measure of classification accuracy that while known, has received relatively little attention. Second, we propose a test for disparate impact based on how well the protected class can be predicted from the other attributes. Third, we describe methods by which data might be made unbiased. Finally, we present empirical evidence supporting the effectiveness of our test for disparate impact and our approach for both masking bias and preserving relevant information in the data. Interestingly, our approach resembles some actual selection practices that have recently received legal scrutiny.

[2]: We study fairness in classification, where individuals are classified, e.g., admitted to a university, and the goal is to prevent discrimination against individuals based on their membership in some group, while maintaining utility for the classifier (the university). The main conceptual contribution of this paper is a framework for fair classification comprising (1) a (hypothetical) task-specific metric for determining the degree to which individuals are similar with respect to the classification task at hand; (2) an algorithm for maximizing utility subject to the fairness constraint, that similar individuals are treated similarly. We also present an adaptation of our approach to achieve the complementary goal of "fair affirmative action," which guarantees statistical parity (i.e., the demographics of the set of individuals receiving any classification are the same as the demographics of the underlying population), while treating similar individuals as similarly as possible. Finally, we discuss the relationship of fairness to privacy: when fairness implies privacy, and how tools developed in the context of differential privacy may be applied to fairness.

[3]: We propose a learning algorithm for fair classification that achieves both group fairness (the proportion of members in a protected group receiving positive classification is identical to the proportion in the population as a whole), and individual fairness (similar individuals should be treated similarly). We formulate fairness as an optimization problem of finding a good representation of the data with two competing goals: to encode the data as well as possible, while simultaneously obfuscating any information about membership in the protected group. We show positive results of our algorithm relative to other known techniques, on three datasets. Moreover, we demonstrate several advantages to our approach. First, our intermediate representation can be used for other classification tasks (i.e., transfer learning is possible); secondly, we take a step toward learning a distance metric which can find important dimensions of the data for classification.

[4]: We study the question of fair clustering under the disparate impact doctrine, where each protected class must have approximately equal representation in every cluster. We formulate the fair clustering problem under both the k-center and the k-median objectives, and show that even with two protected classes the problem is challenging, as the optimum solution can violate common conventions---for instance a point may no longer be assigned to its nearest cluster center! En route we introduce the concept of fairlets, which are minimal sets that satisfy fair representation while approximately preserving the clustering objective. We show that any fair clustering problem can be decomposed into first finding good fairlets, and then using existing machinery for traditional clustering algorithms. While finding good fairlets can be NP-hard, we proceed to obtain efficient approximation algorithms based on minimum cost flow. We empirically demonstrate the by quantifying the value of fair clustering on real-world datasets with sensitive attributes.
With algorithms increasingly running our world, there has been substantial recent interest on incorporating fairness systematically into algorithms and machine learning. One such notion that has gained importance is that of : in addition to requiring that such as gender or race not be used (explicitly) in decisions, this asks that decisions not be disproportionately different for diverse protected classes [1] . This is developed further in the context of clustering in the work of [4] . Such notions of are considered along with -- treating similar individuals similarly -- in [3] . See [2] for earlier work that developed foundations and connections for several such notions of fairness.