Browsing by Author "Dean, Nathaniel"

Now showing 1 - 8 of 8
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    A study of university timetabling that blends graph coloring with the satisfaction of various essential and preferential conditions
    (2004) Redl, Timothy Anton; Dean, Nathaniel; Tapia, Richard A.
    Constructing a satisfactory conflict-free semester-long timetable of courses and creating a similarly satisfactory conflict-free timetable for end-of-semester final examinations are two closely related and often difficult problems that colleges and universities face each semester. We discuss the relevance of such timetabling problems as a natural and practical application of graph coloring, and develop a mathematical and computational model for solving university timetabling problems using techniques of graph coloring that incorporates the satisfaction of both "essential" timetabling conditions (i.e., conditions or constraints that must be satisfied in order to produce a legal or feasible timetable) as well as suggested "preferential" timetabling conditions (i.e., additional conditions or constraints that need not necessarily be satisfied to produce a legal or legitimate timetable, but if satisfied may very well produce a more "acceptable" timetable for students and/or faculty members). We discuss in detail the step-by-step process that is taken to implement our timetabling-by-graph-coloring procedure, from the assembling of university course data, to creating a course conflict graph based on the assembled data, to coloring the conflict graph, to transforming this coloring to a conflict-free timetable, to finally assigning courses to classrooms. Once a conflict-free timetable of courses has been constructed, we present ways in which such a course timetable for a particular semester can be used to construct a conflict-free timetable of final examinations. Our model also considers a number of sociological scheduling concerns and preferences addressed by university registrars, faculty, staff, and students. Computational results, obtained by the author using actual data provided by Rice University and the University of St. Thomas, are documented.
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    Linear-time algorithms for graphs with bounded branchwidth
    (2003) Christian, William Anderson, Jr; Dean, Nathaniel; Cook, William J.
    We present an algorithmic framework (including a single data structure) that is extended into linear-time algorithms to solve several NP-complete graph problems (i.e., INDEPENDENT SET, M AXIMUM CUT, GRAPH COLORING, HAMILTONIAN CYCLE, and DISJOINT PATHS). The linearity is achieved assuming the provision of a branch decomposition of the instance graph. We then modify the framework to create a multithreaded framework that uses the existing problem-specific extensions without any revision. Computational results for the serial and parallel algorithms are provided. In addition, we present a graphical package called JPAD that can display a graph and branch decomposition, show their relationship to each other, and be extended to rim and display the progress and results of algorithms on graphs or on branch decompositions.
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    New algorithms for pathwidth computation
    (2004) Li, Ming; Dean, Nathaniel
    The notions of pathwidth and the closely related treewidth have become more and more important recently. The importance lies not only in theory but also in practice. Theoretically, lots of NP-hard problems become polynomially solvable when restricted in graphs with bounded pathwidth (or treewidth). Practically, pathwidth and treewidth have significant applications in many different fields such as searching games, VLSI design, matrix computation, etc. Computing pathwidth is an NP-complete problem for general graphs, but polynomially solvable for treewidth-bounded graphs. However, there is no known practical algorithm to compute pathwidth for treewidth-bounded graphs. In this dissertation, a new algorithm for computing pathwidth and finding an optimal pathwidth-decomposition for treewidth-bounded graph is presented. This algorithm uses an interval completion technique and the branch-and-bound method to make the pathwidth computation process more efficient, practical, and easy to implement. It can also be easily converted to a parallel algorithm. The data structure for implementing this algorithm is presented, and some computational results are shown. Some heuristic methods to approximate pathwidth for general graphs are also given, especially for series-parallel graphs.
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    On characterizing graphs with branchwidth at most four
    (2001) Riggins, Kymberly Dawn; Dean, Nathaniel
    There are several ways in which we can characterize classes of graphs. One such way of classifying graphs is by their branchwidth. In working to characterize the class of graphs with branchwidth at most four beta 4 we have found a set of reductions that reduces members of beta 4 to the zero graph. We have also computed several planar members of the obstruction set Ob4 for graphs with branchwidth at most four. This thesis will summarize previous results on branchwidth and reveal the previously mentioned new results.
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    On eliminating square paths in a square lattice
    (2000) Williams, Nikki LaTrina; Dean, Nathaniel
    Removing the minimum number of vertices or points from a square lattice such that no square path exists is known as the square path problem. Finding this number as the size of the lattice increases is not so trivial. Results provided by Erdos-Posa and Bienstock-Dean provides an upper bound for eliminating all cycles from a planar graph but sheds little light on the case of the square lattice. This paper provides several values for the minimum number of vertices needed to be removed such that no square path exists.
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    Programming the nanocell, a random array of molecules
    (2002) Husband, Summer Michele; Tour, James M.; Dean, Nathaniel
    The emerging field of molecular electronics seeks to create computational function from individual molecules or arrays of molecules. These nanoscale devices would then enable the production of faster, denser, cheaper computers. Clearly, there are many obstacles to building such devices, one of which is to develop methods for using lithographic wires to address molecules that are many orders of magnitude smaller in size. In this thesis, a moletronics design is presented that offers a method for connecting nanometer molecules to the world-at-large. This architecture involves the production of nanocells, or random arrays of molecules and metallic nanoparticles. The molecules have two discrete states and exhibit electrical behavior that enables complex logic in a nanocell. Methods are presented to take a random array of such switch states and alter them to program a nanocell as a useful logical device. Simulations of this programming process are presented and show that it is theoretically possible to obtain very high level function from these cells. Observations made during simulations are then used to formulate theorems about the programmability of nanocells. These theorems demonstrate that there is a dense solution space of molecular switch states that give rise to certain computation within a nanocell. Future directions of research, such as methods for wiring multiple nanocells together, are included as well.
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    Restricted 2-factors in bipartite graphs
    (2000) Husband, Summer Michele; Dean, Nathaniel
    The k-restricted 2-factor problem is that of finding a spanning subgraph consisting of disjoint cycles with no cycle of length less than or equal to k. It is a generalization of the well known Hamilton cycle problem and is equivalent to this problem when n2≤k≤n-1 . This paper considers necessary and sufficient conditions, algorithms, and polyhedral conditions for 2-factors in bipartite graphs and restricted 2-factors in bipartite graphs. We introduce a generalization of the necessary and sufficient condition for 4-restricted 2-factors in bipartite graphs to 2k-restricted 2-factors in bipartite graphs of a particular form.
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    Survey of results in impartial combinatorial games and an extension to three-player game
    (2001) Venkataraman, Sripriya; Dean, Nathaniel
    There are several known methods to find winning strategies for two-player combinatorial games. This thesis surveys results known for two-player combinatorial games and elucidates a particular winning strategy for a three player game using graph theory. The motivation for work in this area is a belief on the ability of graph theoretic tools to handle multi-player combinatorial game analyses. A winning strategy for two-player undirected vertex Geography game has been formulated earlier and has been shown to be polynomial time. This thesis extends the result to three-player undirected vertex Geography, played on directed trees and answers the question "Does the first player have a winning strategy?". The extension serves as a platform for generalizing results to games involving more than three players.