88 research outputs found
Borel versions of the Local Lemma and LOCAL algorithms for graphs of finite asymptotic separation index
Asymptotic separation index is a parameter that measures how easily a Borel
graph can be approximated by its subgraphs with finite components. In contrast
to the more classical notion of hyperfiniteness, asymptotic separation index is
well-suited for combinatorial applications in the Borel setting. The main
result of this paper is a Borel version of the Lov\'asz Local Lemma -- a
powerful general-purpose tool in probabilistic combinatorics -- under a finite
asymptotic separation index assumption. As a consequence, we show that locally
checkable labeling problems that are solvable by efficient randomized
distributed algorithms admit Borel solutions on bounded degree Borel graphs
with finite asymptotic separation index. From this we derive a number of
corollaries, for example a Borel version of Brooks's theorem for graphs with
finite asymptotic separation index
Probabilistic Programming Interfaces for Random Graphs::Markov Categories, Graphons, and Nominal Sets
We study semantic models of probabilistic programming languages over graphs, and establish a connection to graphons from graph theory and combinatorics. We show that every well-behaved equational theory for our graph probabilistic programming language corresponds to a graphon, and conversely, every graphon arises in this way.We provide three constructions for showing that every graphon arises from an equational theory. The first is an abstract construction, using Markov categories and monoidal indeterminates. The second and third are more concrete. The second is in terms of traditional measure theoretic probability, which covers 'black-and-white' graphons. The third is in terms of probability monads on the nominal sets of Gabbay and Pitts. Specifically, we use a variation of nominal sets induced by the theory of graphs, which covers Erdős-Rényi graphons. In this way, we build new models of graph probabilistic programming from graphons
Equivariant maps to subshifts whose points have small stabilizers
Let be a countably infinite group. Given , we use
to denote the free part of the Bernoulli shift action
of on . Seward and Tucker-Drob showed that there exists a
free subshift such that every
free Borel action of on a Polish space admits a Borel
-equivariant map to . Here we generalize this result as
follows. Let be a subshift of finite type (for example,
could be the set of all proper colorings of the Cayley graph of
with some finite number of colors). Suppose that is a continuous -equivariant
map and let be the set of all group elements that fix
every point in the image of . Unless is constant,
is a finite normal subgroup of . We prove that
there exists a subshift such that the
stabilizer of every point in is and every
free Borel action of on a Polish space admits a Borel
-equivariant map to . As an application, we deduce that
if is a nonempty finite symmetric subset of of size not
containing the identity and is
the set of all proper -colorings of the Cayley graph of
corresponding to , then there is a free subshift such that every free Borel action of on a Polish
space admits a Borel -equivariant map to .Comment: 22 p
Distributed Algorithms, the Lov\'{a}sz Local Lemma, and Descriptive Combinatorics
In this paper we consider coloring problems on graphs and other combinatorial
structures on standard Borel spaces. Our goal is to obtain sufficient
conditions under which such colorings can be made well-behaved in the sense of
topology or measure. To this end, we show that such well-behaved colorings can
be produced using certain powerful techniques from finite combinatorics and
computer science. First, we prove that efficient distributed coloring
algorithms (on finite graphs) yield well-behaved colorings of Borel graphs of
bounded degree; roughly speaking, deterministic algorithms produce Borel
colorings, while randomized algorithms give measurable and Baire-measurable
colorings. Second, we establish measurable and Baire-measurable versions of the
Symmetric Lov\'{a}sz Local Lemma (under the assumption
, which is stronger than the standard
LLL assumption but still sufficient
for many applications). From these general results, we derive a number of
consequences in descriptive combinatorics and ergodic theory.Comment: 35 page
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