T. Ryan Hoens


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

Publication Date


Document Type


Department, Program, or Center

Computer Science (GCCIS)


Radziszowski, Stanisław

Advisor/Committee Member

Faliszewski, Piotr


Note: imported from RIT’s Digital Media Library running on DSpace to RIT Scholar Works. Physical copy available through RIT's The Wallace Library at: QA166.22 .H64 2008


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