  Clustering is a fundamental problem in unsupervised machine learning, and
fair variants of it have recently received significant attention due to its
societal implications. In this work we introduce a novel definition of
individual fairness for clustering problems. Specifically, in our model, each
point $j$ has a set of other points $\mathcal{S}_j$ that it perceives as
similar to itself, and it feels that it is fairly treated if the quality of
service it receives in the solution is $\alpha$-close (in a multiplicative
sense, for a given $\alpha \geq 1$) to that of the points in $\mathcal{S}_j$.
We begin our study by answering questions regarding the structure of the
problem, namely for what values of $\alpha$ the problem is well-defined, and
what the behavior of the \emph{Price of Fairness (PoF)} for it is. For the
well-defined region of $\alpha$, we provide efficient and easily-implementable
approximation algorithms for the $k$-center objective, which in certain cases
enjoy bounded-PoF guarantees. We finally complement our analysis by an
extensive suite of experiments that validates the effectiveness of our
theoretical results.
