Combinatorial optimization and unsupervised machine learning problems have been extensively studied and are relatively well-understood. Examples of such problems that play a central role in this work are clustering problems and problems of finding cuts in graphs. The goal of the research presented in this dissertation is to introduce novel variants of the aforementioned problems, by generalizing their classic variants into two, not necessarily disjoint, directions. The first direction involves incorporating fairness aspects to a problem's specifications, and the second involves the introduction of some form of randomness in the problem definition, e.g., stochastic uncertainty about the problem's parameters.

Fairness in the design of algorithms and in machine learning has received a significant amount of attention during the last few years, mainly due to the realization that standard optimization approaches can frequently lead to severely unfair outcomes, that can potentially hurt the individuals or the groups involved in the corresponding application. As far as considerations of fairness are concerned, in this work we begin by presenting two novel individually-fair clustering models, together with algorithms with provable guarantees for them. The first such model exploits randomness in order to provide fair solutions, while the second is purely deterministic. The high-level motivation behind both of them is trying to treat similar individuals similarly. Moving forward, we focus on a graph cut problem that captures situations of disaster containment in a network. For this problem we introduce two novel fair variants. The first variant focuses on demographic fairness, while the second considers a probabilistic notion of individual fairness. Again, we give algorithms with provable guarantees for the newly introduced variants.

In the next part of this thesis we turn our attention to generalizing problems through the introduction of stochasticity. At first, we present algorithmic results for a computational epidemiology problem, whose goal is to control the stochastic diffusion of a disease in a contact network. This problem can be interpreted as a stochastic generalization of a static graph cut problem. Finally, this dissertation also includes work on a well-known paradigm in stochastic optimization, namely the two-stage stochastic setting with recourse. Two-stage problems capture a wide variety of applications revolving around the trade-off between provisioning and rapid response. In this setting, we present a family of clustering problems that had not yet been studied in the literature, and for this family we show novel algorithmic techniques that provide constant factor approximation algorithms.

We conclude the dissertation with a discussion on open problems and future research directions in the general area of algorithmic fairness.