In this thesis defense, we will discuss the following results.

Over the last three decades, the online bipartite matching (OBM) problem has emerged as a central problem in the area of Online Algorithms. Perhaps even more important is its role in the area of Matching-Based Market Design.

Algebraic Complexity Theory is a field in which one studies complexity theoretic questions surrounding algebraic objects. In this talk we will be broadly discussing two such problems.

Many low- and middle-income countries face limited supply of vaccines. In such situations it is imperative to devise vaccination rollout strategies that maximize the cost-effectiveness of these limited vaccine stocks.