Hicks, Illya V2024-05-212024-05-212024-052024-04-16May 2024Kroger, Samuel. Clique Relaxations & the Minority Districting Problem. (2024). PhD diss., Rice University. https://hdl.handle.net/1911/116106https://hdl.handle.net/1911/116106This thesis studies the intersection of graph theory and mixed integer programming through three combinatorial optimization problems. We model each problem on graphs and exploit the inherent structure of the problem to propose novel integer programming formulations, create fixing procedures, and add valid cuts to the problem. We demonstrate the theoretical and computational improvement for each problem through propositions and computational experiments. The Anchored k-core Problem is a variant of the maximum k-core problem, which itself is a relaxation of a clique. We propose an integer programming formulation for the anchored k-core problem, define and study the polyhedra of the maximum anchored k-core problem, and provide a computational study comparing our model against leading algorithms in the literature. Next, we cover the Maximum Stable Set Problem, which is closely related to the maximum clique problem. We extend a polynomial time algorithm for solving the maximum stable set problem on chordal graphs into a polynomial time fixing procedure for general graphs. We also discover a small class of graphs for which the maximum stable set problem is polynomial-time solvable using our proposed algorithm. Finally, we cover the Minority Districting Problem. In this problem, we hope to identify states with a legal impetus, imposed by Section 2 of the Voting Rights Act,to form minority-majority districts. Identifying and enacting plans with minority districts is paramount to ensure minority populations in America are given the political representation they are constitutionally entitled to. We use a diameter-based metric to enforce compactness based on s-clubs (another clique relaxation). We propose a new mixed integer programming formulation alongside robust fixing procedures, symmetry-breaking constraints, and a framework for finding the maximum number of minority districts possible for a state with a diameter-bounded compactness measure.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.Operations ResearchGraph TheoryInteger ProgrammingCombinatorial OptimizationThesis2024-05-21