Since Euler began studying paths in graphs, graph theory has become an important branch of mathematics. With all of the research into graph theoretic problems, however, counting – exactly or approximately – the number of simple paths in finite graphs has been neglected. This thesis investigates an approximation technique known as Markov chain Monte Carlo (MCMC) for the specific purpose of approximating the number of simple paths in graphs. Due to the paucity of research into the subject, the thesis will make the conjecture that this cannot be done exactly in an efficient manner (assuming that the longstanding conjecture P 6= NP holds). To this end, the thesis focuses on the relationship between counting and sampling in both weighted and unweighted complete graphs, trees, and directed acyclic graphs (DAGs). This includes both positive and negative results for sampling, as well as demonstrating how counting and sampling are intimately related problems.
Library of Congress Subject Headings
Paths and cycles (Graph theory); Graph theory--Data processing; Monte Carlo method; Markov processes
Department, Program, or Center
Computer Science (GCCIS)
Hoens, T. Ryan, "Counting and sampling paths in graphs" (2008). Thesis. Rochester Institute of Technology. Accessed from
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