Sampling of combinatorial structures is an important statistical tool used in applications in a number of areas ranging from statistical physics, data mining, to biological sciences. Of comparable importance is the computation of the cor- responding partition function, which, in the case of the uniform distribution, is equivalent to the problem of counting all such structures. For self-reducible combinatorial structures, once we can produce an almost uniform sample from them, then we can approximately count them.
Using a Markov chain Monte Carlo method, this thesis presents polynomial-time algorithms to approximately count and almost uniformly sample crossing-free matchings for certain input classes of graphs. Since the problem in its generality appears to be difficult, we made natural restrictions on the in- put graphs. Namely, we consider vertices arranged in a grid in the plane, where edges are line segments connecting the vertices and a matching is crossing-free if no two matching edges intersect. For appropriate bounds on the dimensions of the grid and the edge lengths, we show that a natural Markov chain is rapidly mixing and that the problem is self-reducible.
Library of Congress Subject Headings
Combinatorial analysis; Markov processes; Graph theory
Computer Science (MS)
Department, Program, or Center
Computer Science (GCCIS)
Stanisław P. Radziszowski
Sun, Wenbo, "Sampling and Counting Crossing-Free Matchings" (2016). Thesis. Rochester Institute of Technology. Accessed from
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