Description
- Abstract:
- The k-max-cut problem is a combinatorial question of great importance in network science and data science. Given a network, the problem seeks a way to cut the network into k pieces in a way that maximizes the value of the cut edges. This has immediate applications to any setting where a large collection of items is accompanied by some notion of similarity or connectivity, as the solution to the problem should yield groups of items that are highly similar or connected to each other. For example, many data clustering problems can be examined as max-cut problems. Here, we consider the problem from two angles. We first examine the solution space of the max-cut problem under a relaxation that represents the discrete combinatorial solutions in a continuous geometric body, identified as the up-to-k-partition polytope. We give several results on the edge structure and low-dimensional faces of up-to-k-partition polytopes, in particular finding that many of them recursively possess smaller up-to-k-partition polytopes as faces. Next we evaluate a semidefinite programming algorithm for solving the max-cut problem in the specific context of data clustering, comparing its performance with well-established clustering algorithms. These experimental simulations use both randomly-generated data and random samples from a high-dimensional image dataset to challenge the clustering algorithms, with the results showing that the semidefinite programming approach yields sharper indications of the correct number of groups in the dataset. We also include two independent projects which are distinct from the k-max-cut problem but are linked to the dissertation through the common lens of combinatorics. The first of these is a project on the combinatorics and geometry of polytopes representing generalized parking functions, which characterizes the behavior and properties of the polytopes in the broad context of polytope theory. The second examines arboricity polynomials, a graph invariant determined by a subset of the partitions of graph edges, and finds that the partitions are best represented by considering the polynomials in a falling-factorial basis rather than the standard basis.
- Notes:
- Thesis (Ph. D.)--Brown University, 2025
Citation
Chambers, Teressa,
"Geometric, Combinatorial, and Experimental Perspectives on the k-Max-Cut Problem"
(2025).
Applied Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://repository.library.brown.edu/studio/item/bdr:6zpc7xvu/
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Applied Mathematics Theses and Dissertations
Theses and Dissertations for the Applied Mathematics department....