Two Problems in Discrete Mathematics

• Main Author: Chiarelli, John
• Format: Dissertation
• Language: English
• Published: ProQuest Dissertations & Theses 01.01.2020
• Subjects: Mathematics
• ISBN: 9798557093286

This thesis centers around two projects that I have undertaken in the subject of discrete mathematics. The primary project pertains to the stable matching problem, and puts particular focus on a relaxation of stability that we call S-stability. The secondary project looks at boolean functions as polynomials, and seeks to understand and use a complexity measure called the maxonomial hitting set size.The stable matching problem is a well-known problem in discrete mathematics, with many practical applications for the algorithms derived from it. Our investigations into the stable matching problem center around the operation ψ : E(G(I)) → E(G(I)); we show that for sufficiently large k, ψIk maps everything to a set of edges that we call the hub, and give algorithms for evaluating ψI(S) for specific values of S. Subsequently, we extend results on the lattice structure of stable matchings to S-stability and consider the polytope of fractional matchings for these same weaker notions of stability. We also reflect on graphs represented by instances with every edge in the hub.Given a boolean function f : {0, 1}n → {0, 1}, it is well-known that it can be represented as a unique multilinear polynomial. We improve a result by Nisan and Szegedy on the maximum number of relevant variables in a low degree boolean polynomial using the maxonomial hitting set size, and look at the largest possible maxonomial hitting set size for a degree d boolean function.