Many modern problems in algorithms and optimization are driven by data which often carries with it an element of uncertainty. In this work, we conduct an investigation into algorithmic foundations and applications across three main areas.

The first area is online matching algorithms for e-commerce applications such as online sales and advertising. The importance of e-commerce in modern business cannot be overstated and even minor algorithmic improvements can have huge impacts. In online matching problems, we generally have a known offline set of goods or advertisements while users arrive online and allocations must be made immediately and irrevocably when a user arrives. However, in the real world, there is also uncertainty about a user's true interests and this can be modeled by considering matching problems in a graph with stochastic edges that only have a probability of existing. These edges can represent the probability of a user purchasing a product or clicking on an ad. Thus, we optimize over data which only provides an estimate of what types of users will arrive and what they will prefer. We survey a broad landscape of problems in this area, gain a deeper understanding of the algorithmic challenges, and present algorithms with improved worst case performance

The second area is constrained clustering where we explore classical clustering problems with additional constraints on which data points should be clustered together. Utilizing these constraints is important for many clustering problems because they can be used to ensure fairness, exploit expert advice, or capture natural properties of the data. In simplest case, this can mean some pairs of points have ``must-link'' constraints requiring that that they must be clustered together. Moving into stochastic settings, we can describe more general pairwise constraints such as bounding the probability that two points are separated into different clusters. This lets us introduce a new notion of fairness for clustering and address stochastic problems such as semi-supervised learning with advice from imperfect experts. Here, we introduce new models of constrained clustering including new notions of fairness for clustering applications. Since these problems are NP-hard, we give approximation algorithms and in some cases conduct experiments to explore how the algorithms perform in practice. Finally, we look closely at the particular clustering problem of drawing election districts and show how constraining the clusters based on past voting data can interact with voter incentives.

The third area is string algorithms for bioinformatics and metagenomics specifically where the data deluge from next generation sequencing drives the necessity for new algorithms that are both fast and accurate. For metagenomic analysis, we present a tool for clustering a microbial marker gene, the 16S ribosomal RNA gene. On the more theoretical side, we present a succinct application of the Method of the Four Russians to edit distance computation as well as new algorithms and bounds for the maximum duo-preservation string mapping (MPSM) problem.