68,208 research outputs found
Determining full conditional independence by low-order conditioning
A concentration graph associated with a random vector is an undirected graph
where each vertex corresponds to one random variable in the vector. The absence
of an edge between any pair of vertices (or variables) is equivalent to full
conditional independence between these two variables given all the other
variables. In the multivariate Gaussian case, the absence of an edge
corresponds to a zero coefficient in the precision matrix, which is the inverse
of the covariance matrix. It is well known that this concentration graph
represents some of the conditional independencies in the distribution of the
associated random vector. These conditional independencies correspond to the
"separations" or absence of edges in that graph. In this paper we assume that
there are no other independencies present in the probability distribution than
those represented by the graph. This property is called the perfect
Markovianity of the probability distribution with respect to the associated
concentration graph. We prove in this paper that this particular concentration
graph, the one associated with a perfect Markov distribution, can be determined
by only conditioning on a limited number of variables. We demonstrate that this
number is equal to the maximum size of the minimal separators in the
concentration graph.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ193 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Algorithms and Bounds for Very Strong Rainbow Coloring
A well-studied coloring problem is to assign colors to the edges of a graph
so that, for every pair of vertices, all edges of at least one shortest
path between them receive different colors. The minimum number of colors
necessary in such a coloring is the strong rainbow connection number
(\src(G)) of the graph. When proving upper bounds on \src(G), it is natural
to prove that a coloring exists where, for \emph{every} shortest path between
every pair of vertices in the graph, all edges of the path receive different
colors. Therefore, we introduce and formally define this more restricted edge
coloring number, which we call \emph{very strong rainbow connection number}
(\vsrc(G)).
In this paper, we give upper bounds on \vsrc(G) for several graph classes,
some of which are tight. These immediately imply new upper bounds on \src(G)
for these classes, showing that the study of \vsrc(G) enables meaningful
progress on bounding \src(G). Then we study the complexity of the problem to
compute \vsrc(G), particularly for graphs of bounded treewidth, and show this
is an interesting problem in its own right. We prove that \vsrc(G) can be
computed in polynomial time on cactus graphs; in contrast, this question is
still open for \src(G). We also observe that deciding whether \vsrc(G) = k
is fixed-parameter tractable in and the treewidth of . Finally, on
general graphs, we prove that there is no polynomial-time algorithm to decide
whether \vsrc(G) \leq 3 nor to approximate \vsrc(G) within a factor
, unless PNP
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