Clique Generalizations and Related Problems

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dc.contributor.advisorHicks, Illya Ven_US
dc.creatorWood, Cynthia Ivetteen_US
dc.date.accessioned2017-08-02T16:20:15Zen_US
dc.date.available2017-08-02T16:20:15Zen_US
dc.date.created2016-05en_US
dc.date.issued2016-04-21en_US
dc.date.submittedMay 2016en_US
dc.date.updated2017-08-02T16:20:15Zen_US
dc.description.abstractA large number of real-world problems can be model as optimization problems in graphs. The clique model was introduced to aid the study of network structure for social interaction. Each vertex represented an actor and the edges represented the relations between them. Nevertheless, the model has been shown to be restrictive for modeling real-world problems, since it leaves out subgraphs that do not have all pos- sible edges. As a consequence, clique generalizations were introduced to overcome the disadvantages of the clique model. In this thesis, I present three computationally dif- ficult combinatorial optimization problems related to clique generalization problems: co-2-plexes and k-cores. A k-core is a subgraph with minimum degree greater than or equal to k. In this work, I discuss the minimal k-core problem and the minimum k-core problem. I present a backtracking algorithm to find all minimal k-cores of a given undirected graph and its applications to the study of associative memory. The proposed method is a modification of the Bron and Kerbosch algorithm for finding all cliques of an undirected graph. In addition, I study the polyhedral structure of the k-core polytope. The minimum k-core problem is modeled as a binary integer program and relaxed as a linear program. Since the relaxation yields to a non-integral solution, cuts must be added in order to improve the solution. I show that edge and cycle transversals of the graph give valid inequalities for the convex hull of k-cores. A set of pairwise non-adjacent vertices defines a stable set. A stable set is the complement of a clique. A co-2-plex is a subgraph with degree less than or equal to one, and it is a stable set relaxation. I introduce a study of the maximum weighted co-2-plex (MWC2P) problem for {claw, bull}-free graphs and present two polynomial time algorithms to solve it. One of the algorithms transforms the original graph to solve an instance of the maximum weighted stable set problem utilizing Minty’s algorithm. The second algorithm is an extension of Minty’s algorithm and solves the problem in the original graph. All the algorithms discussed in this thesis were implemented and tested. Numerical results are provided for each one of them.en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.citationWood, Cynthia Ivette. "Clique Generalizations and Related Problems." (2016) Diss., Rice University. <a href="https://hdl.handle.net/1911/96187">https://hdl.handle.net/1911/96187</a>.en_US
dc.identifier.urihttps://hdl.handle.net/1911/96187en_US
dc.language.isoengen_US
dc.rightsCopyright 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.en_US
dc.subjectCliquesen_US
dc.subjectClique Generalizationsen_US
dc.subjectk-coreen_US
dc.subjectminimum k-coreen_US
dc.subjectminimal k-coreen_US
dc.subjectk-plexen_US
dc.subjectco-k-plexen_US
dc.subjectcell assemblyen_US
dc.subjectk-assemblyen_US
dc.subjectmemoryen_US
dc.subjectgraph analyticsen_US
dc.subjectgraph Algorithmsen_US
dc.titleClique Generalizations and Related Problemsen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentComputational and Applied Mathematicsen_US
thesis.degree.disciplineEngineeringen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophyen_US
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