R.Moser, G.Tardos: A constructive proof of the general Lovász Local Lemma (given by: Marek Krčál)
(prob. space is
product of coordinates)
coordinate in
set of events in
each
depends on
some unique minimal
subset
of
coordinates
dependency graph of in
the sense of Moser:
form an edge
iff
intersects
+ neighborhood
+
of
in
(including
)
We need to formalize the random source of the algorithm: an infinite sequence of random samples for each
coordinate
, i.e. an element of
. Having the input of the algorithm fixed, we get the
following function
example:
:
C
:
:
log of execution:
the algorithm.
is sequence of resampled events in their order in execution of
witness tree:
is a finite rooted tree with labeling
that children of a vertex
receive labels from
of its vertices such
.
a witness tree associated with resampling step . It is defined as
, which we get backward-
inductively as follows:
is defined as isolated root labeled
construct from
.
: select any
distance from the root. Attach a new child
.
such that
labelled
with maximal
to the vertex . If no such
If distinct children of the same vertex receive distinct labels, we call the tree proper.
the set of all proper witness trees having the root labelled by
Outline of the proof:
Galton-Watson random process:
.
exists,