88 research outputs found

    Borel versions of the Local Lemma and LOCAL algorithms for graphs of finite asymptotic separation index

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    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

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    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

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    Let Γ\Gamma be a countably infinite group. Given kNk \in \mathbb{N}, we use Free(kΓ)\mathrm{Free}(k^\Gamma) to denote the free part of the Bernoulli shift action of Γ\Gamma on kΓk^\Gamma. Seward and Tucker-Drob showed that there exists a free subshift SFree(2Γ)\mathcal{S} \subseteq \mathrm{Free}(2^\Gamma) such that every free Borel action of Γ\Gamma on a Polish space admits a Borel Γ\Gamma-equivariant map to S\mathcal{S}. Here we generalize this result as follows. Let S\mathcal{S} be a subshift of finite type (for example, S\mathcal{S} could be the set of all proper colorings of the Cayley graph of Γ\Gamma with some finite number of colors). Suppose that π ⁣:Free(kΓ)S\pi \colon \mathrm{Free}(k^\Gamma) \to \mathcal{S} is a continuous Γ\Gamma-equivariant map and let Stab(π)\mathrm{Stab}(\pi) be the set of all group elements that fix every point in the image of π\pi. Unless π\pi is constant, Stab(π)\mathrm{Stab}(\pi) is a finite normal subgroup of Γ\Gamma. We prove that there exists a subshift SS\mathcal{S}' \subseteq \mathcal{S} such that the stabilizer of every point in S\mathcal{S}' is Stab(π)\mathrm{Stab}(\pi) and every free Borel action of Γ\Gamma on a Polish space admits a Borel Γ\Gamma-equivariant map to S\mathcal{S}'. As an application, we deduce that if FF is a nonempty finite symmetric subset of Γ\Gamma of size F=d|F| = d not containing the identity and Col(F,d+1)(d+1)Γ\mathrm{Col}(F, d + 1) \subseteq (d+1)^\Gamma is the set of all proper (d+1)(d+1)-colorings of the Cayley graph of Γ\Gamma corresponding to FF, then there is a free subshift SCol(F,d+1)\mathcal{S} \subseteq \mathrm{Col}(F, d+1) such that every free Borel action of Γ\Gamma on a Polish space admits a Borel Γ\Gamma-equivariant map to S\mathcal{S}.Comment: 22 p

    Distributed Algorithms, the Lov\'{a}sz Local Lemma, and Descriptive Combinatorics

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    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 p(d+1)8215\mathsf{p}(\mathsf{d}+1)^8 \leq 2^{-15}, which is stronger than the standard LLL assumption p(d+1)e1\mathsf{p}(\mathsf{d} + 1) \leq e^{-1} 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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