Browsing by Author "Kroger, Samuel"
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Item Clique Relaxations & the Minority Districting Problem(2024-04-16) Kroger, Samuel; Hicks, Illya VThis 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.Item The Maximum Anchored k-core Problem: Mixed Integer Programming Formulations(2022-04-14) Kroger, Samuel; Hicks, Illya VThe maximum anchored k-core problem plays an important role in marketing, network architecture, and social media; the problem allows network designers and influencers to find the most pivotal vertices which increase the size of the network. In this thesis, we investigate two mixed integer programming (MIP) formulations for the maximum anchored k-core problem: (i) a naive model and (ii) a strong model. We examine the MIP formulations analytically and computationally. We also compare the computational performance of the MIP models with two existing heuristic algorithms: Residual Core Maximization (RCM) and Onion-Layer based Anchored k-core (OLAK). Furthermore, we propose valid inequalities and fixing procedures to improve the computational performance of the MIP models. Finally, we conduct experiments on a set of benchmark instances. Our computational experiments show the superiority of the strong model against the naive model, and the heuristics.