41 research outputs found
Metric Construction, Stopping Times and Path Coupling
In this paper we examine the importance of the choice of metric in path
coupling, and the relationship of this to \emph{stopping time analysis}. We
give strong evidence that stopping time analysis is no more powerful than
standard path coupling. In particular, we prove a stronger theorem for path
coupling with stopping times, using a metric which allows us to restrict
analysis to standard one-step path coupling. This approach provides insight for
the design of non-standard metrics giving improvements in the analysis of
specific problems.
We give illustrative applications to hypergraph independent sets and SAT
instances, hypergraph colourings and colourings of bipartite graphs.Comment: 21 pages, revised version includes statement and proof of general
stopping times theorem (section 2.2), and additonal remarks in section
Path Coupling Using Stopping Times and Counting Independent Sets and Colourings in Hypergraphs
We give a new method for analysing the mixing time of a Markov chain using
path coupling with stopping times. We apply this approach to two hypergraph
problems. We show that the Glauber dynamics for independent sets in a
hypergraph mixes rapidly as long as the maximum degree Delta of a vertex and
the minimum size m of an edge satisfy m>= 2Delta+1. We also show that the
Glauber dynamics for proper q-colourings of a hypergraph mixes rapidly if m>= 4
and q > Delta, and if m=3 and q>=1.65Delta. We give related results on the
hardness of exact and approximate counting for both problems.Comment: Simpler proof of main theorem. Improved bound on mixing time. 19
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Quantifying the difference between phylogenetic diversity and diversity indices
Phylogenetic diversity is a popular measure for quantifying the biodiversity of a collection Y of species, while phylogenetic diversity indices provide a way to apportion phylogenetic diversity to individual species. Typically, for some specific diversity index, the phylogenetic diversity of Y is not equal to the sum of the diversity indices of the species in Y. In this paper, we investigate the extent of this difference for two commonly-used indices: Fair Proportion and Equal Splits. In particular, we determine the maximum value of this difference under various instances including when the associated rooted phylogenetic tree is allowed to vary across all rooted phylogenetic trees with the same leaf set and whose edge lengths are constrained by either their total sum or their maximum value
Quantifying the difference between phylogenetic diversity and diversity indices
Phylogenetic diversity is a popular measure for quantifying the biodiversity
of a collection of species, while phylogenetic diversity indices provide a
way to apportion phylogenetic diversity to individual species. Typically, for
some specific diversity index, the phylogenetic diversity of is not equal
to the sum of the diversity indices of the species in In this paper, we
investigate the extent of this difference for two commonly-used indices: Fair
Proportion and Equal Splits. In particular, we determine the maximum value of
this difference under various instances including when the associated rooted
phylogenetic tree is allowed to vary across all root phylogenetic trees with
the same leaf set and whose edge lengths are constrained by either their total
sum or their maximum value.Comment: 26 pages, 5 figure
Computing the hybridization number is fixed-parameter tractable
Reticulation processes in evolution mean that the ancestral history of certain groups of present-day species is non-tree-like. These processes include hybridization, lateral gene transfer, and recombination. Despite the existence of reticulation, such events are relatively rare and so a fundamental problem for biologists is the following: given a collection of rooted binary phylogenetic trees on sets of species that correctly represent the tree-like evolution of different parts of their genomes, what is the smallest number of "reticulation" vertices in any network that explains the evolution of the species under consideration. It has been previously shown that this problem is NP-hard even when the collection consists of only two rooted binary phylogenetic trees; however, in this paper, we show that it is fixed-parameter tractable