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

Publication Date


Document Type


Student Type


Degree Name

Computer Science (MS)

Department, Program, or Center

Computer Science (GCCIS)


Ivona Bezakova

Advisor/Committee Member

Stanisław P. Radziszowski

Advisor/Committee Member

Christopher Homan


Physical copy available from RIT's Wallace Library at QA166.243 .S86 2016


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