The research talk that I gave at the Joint Mathematics Meetings has now been posted to my talks webpage. This talk was about the research that one of my groups accomplished this last summer during the Claremont REU.
Title Using TPA to count linear extensions
Authors Jacqueline Banks, Scott Garrabrant, Mark L Huber, and Anne Perizzolo
Abstract A linear extension of a poset P is a permutation of the elements of the set that respects the partial order. Let L(P) denotethe number of linear extensions. It is a #P complete problem to determine L(P) exactly for an arbitrary poset, and so randomized approximation algorithms that draw randomly from the set of linear extensions are used. In this work, the set of linear extensions is embedded in a larger state space with a continuous parameter. The introduction of a continuous parameter allows for the use of a more efficient method for approximating L(P) called TPA. Our primary result is that it is possible to sample from this continuous embedding in time that as fast or faster than the best known methods for sampling uniformly from linear extensions. For a poset containing n elements, this means we can approximate L(P) towithin a factor of 1 + epsilon with probability at least 1 – delta using an expected number of random bits and comparisons in the poset which is at most O(n^3(ln n)(ln L(P))^2).