Instance selection (or prototype selection) [García et al.(2015)] is a preprocessing task in machine learning (or data mining) that aims at selecting a subset of the data instances composing the training set that a machine learning algorithm will use. There are two main reasons to perform this task: efficiency and cleaning. Reducing the size of the training set reduces the computational cost of running the machine learning algorithm, especially in the case of instance-based classifiers like KNN (see the Preliminaries section for a description of KNN classifiers). Furthermore, we may be interested in removing noisy instances from the training set: instances due to errors or other causes can induce mistakes in the machine learning algorithm.

The goal of multi-winner elections is to choose a fixed-size committee based on voters’ preferences. An important concern in this setting is representation: large groups of voters with cohesive preferences should be adequately represented by the election winners. Recently, Aziz et al. proposed two axioms that aim to capture this idea: justified representation (JR) and its strengthening extended justified representation (EJR). In this paper, we extend the work of Aziz et al. in several directions. First, we answer an open question of Aziz et al., by showing that Reweighted Approval Voting satisfies JR for k = 3; 4; 5, but fails it for k >= 6. Second, we observe that EJR is incompatible with the Perfect Representation criterion, which is important for many applications of multi-winner voting, and propose a relaxation of EJR, which we call Proportional Justified Representation (PJR). PJR is more demanding than JR, but, unlike EJR, it is compatible with perfect representation, and a committee that provides PJR can be computed in polynomial time if the committee size divides the number of voters. Moreover, just like EJR, PJR can be used to characterize the classic PAV rule in the class of weighted PAV rules. On the other hand, we show that EJR provides stronger guarantees with respect to average voter satisfaction than PJR does.