4,417 research outputs found
Complete Acyclic Colorings
We study two parameters that arise from the dichromatic number and the
vertex-arboricity in the same way that the achromatic number comes from the
chromatic number. The adichromatic number of a digraph is the largest number of
colors its vertices can be colored with such that every color induces an
acyclic subdigraph but merging any two colors yields a monochromatic directed
cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest
number of colors that can be used such that every color induces a forest but
merging any two yields a monochromatic cycle. We study the relation between
these parameters and their behavior with respect to other classical parameters
such as degeneracy and most importantly feedback vertex sets.Comment: 17 pages, no figure
Partition MCMC for inference on acyclic digraphs
Acyclic digraphs are the underlying representation of Bayesian networks, a
widely used class of probabilistic graphical models. Learning the underlying
graph from data is a way of gaining insights about the structural properties of
a domain. Structure learning forms one of the inference challenges of
statistical graphical models.
MCMC methods, notably structure MCMC, to sample graphs from the posterior
distribution given the data are probably the only viable option for Bayesian
model averaging. Score modularity and restrictions on the number of parents of
each node allow the graphs to be grouped into larger collections, which can be
scored as a whole to improve the chain's convergence. Current examples of
algorithms taking advantage of grouping are the biased order MCMC, which acts
on the alternative space of permuted triangular matrices, and non ergodic edge
reversal moves.
Here we propose a novel algorithm, which employs the underlying combinatorial
structure of DAGs to define a new grouping. As a result convergence is improved
compared to structure MCMC, while still retaining the property of producing an
unbiased sample. Finally the method can be combined with edge reversal moves to
improve the sampler further.Comment: Revised version. 34 pages, 16 figures. R code available at
https://github.com/annlia/partitionMCM
Digraph Complexity Measures and Applications in Formal Language Theory
We investigate structural complexity measures on digraphs, in particular the
cycle rank. This concept is intimately related to a classical topic in formal
language theory, namely the star height of regular languages. We explore this
connection, and obtain several new algorithmic insights regarding both cycle
rank and star height. Among other results, we show that computing the cycle
rank is NP-complete, even for sparse digraphs of maximum outdegree 2.
Notwithstanding, we provide both a polynomial-time approximation algorithm and
an exponential-time exact algorithm for this problem. The former algorithm
yields an O((log n)^(3/2))- approximation in polynomial time, whereas the
latter yields the optimum solution, and runs in time and space O*(1.9129^n) on
digraphs of maximum outdegree at most two. Regarding the star height problem,
we identify a subclass of the regular languages for which we can precisely
determine the computational complexity of the star height problem. Namely, the
star height problem for bideterministic languages is NP-complete, and this
holds already for binary alphabets. Then we translate the algorithmic results
concerning cycle rank to the bideterministic star height problem, thus giving a
polynomial-time approximation as well as a reasonably fast exact exponential
algorithm for bideterministic star height.Comment: 19 pages, 1 figur
Parameterized Algorithms for Directed Maximum Leaf Problems
We prove that finding a rooted subtree with at least leaves in a digraph
is a fixed parameter tractable problem. A similar result holds for finding
rooted spanning trees with many leaves in digraphs from a wide family
that includes all strong and acyclic digraphs. This settles completely an open
question of Fellows and solves another one for digraphs in . Our
algorithms are based on the following combinatorial result which can be viewed
as a generalization of many results for a `spanning tree with many leaves' in
the undirected case, and which is interesting on its own: If a digraph of order with minimum in-degree at least 3 contains a rooted
spanning tree, then contains one with at least leaves
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