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

@article{Bernshteyn2020DistributedAT, title={Distributed Algorithms, the Lov\'\{a\}sz Local Lemma, and Descriptive Combinatorics}, author={Anton Bernshteyn}, journal={arXiv: Combinatorics}, year={2020} }

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…

## Tables from this paper

## 15 Citations

### Descriptive Combinatorics, Computable Combinatorics, and ASI Algorithms

- Mathematics, Computer Science
- 2022

New types of local algorithms are introduced, which are called “ASI Algorithms”, and used to demonstrate a link between descriptive and computable combinatorics.

### A Cantor--Bendixson dichotomy of domatic partitions

- Mathematics
- 2022

Let Γ = ∏ i ∈ N Γ i be an inﬁnite product of nontrivial ﬁnite groups, or let Γ = ( R / Z ) n be a ﬁnite-dimensional torus. For a countably inﬁnite set S ⊆ Γ , an ℵ 0 -domatic partition is a partial ℵ…

### Deterministic Distributed algorithms and Descriptive Combinatorics on Δ-regular trees

- Mathematics, Computer ScienceArXiv
- 2022

It is shown that a local problem admits a continuous solution if and only if it admits a local algorithm with local complexity O (log ∗ n ) , and a Baire measurable solution is admitted if andonly if it admitting a local algorithms with local simplicity O ( log n ) .

### Deterministic Distributed algorithms and Descriptive Combinatorics on \Delta-regular trees

- Mathematics, Computer Science
- 2022

It is shown that a local problem admits a continuous solution if and only if it admits a local algorithm with local complexity O (log ∗ n ) , and a Baire measurable solution is admitted if andonly if it admitting a local algorithms with local simplicity O ( log n ) .

### Moser-Tardos Algorithm with small number of random bits

- Mathematics, Computer ScienceArXiv
- 2022

A deterministic algorithm for finding a satisfying assignment, which in any class of problems as in the previous paragraph runs in time O(n), where n is the number of variables.

### The Landscape of Distributed Complexities on Trees and Beyond

- Computer Science, MathematicsPODC
- 2022

The main contribution is to complete the classification of the complexity landscape of LCL problems on trees in the LOCAL model, by proving that every LCL problem with local complexity o (log* n) has actually complexityO(1), which improves upon the previous speedup result.

### Local Problems on Trees from the Perspectives of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics

- MathematicsITCS
- 2022

This approach that borrows techniques from the fields (a), (b) and (c) implies a number of results about possible complexities of finitary factor solutions and helps to view all three perspectives as a part of a common theory of locality.

### A Fast Distributed Algorithm for (Δ+1)-Edge-Coloring

- Mathematics, Computer ScienceJ. Comb. Theory, Ser. B
- 2022

### Classification of Local Problems on Paths from the Perspective of Descriptive Combinatorics

- MathematicsArXiv
- 2021

It is known that randomness does not help with solving local problems on oriented paths and there are four classes of local problems and most classes have natural definitions in all three fields.

### Of Toasts and Tails

- MathematicsArXiv
- 2021

This work presents an intimate connection among the following fields: distributed local algorithms, finitary factors of iid processes, and descriptive combinatorics: coming from the area of computer science, to view all three perspectives as a part of a common theory of locality.

## References

SHOWING 1-10 OF 49 REFERENCES

### A Note on Vertex List Colouring

- MathematicsCombinatorics, Probability and Computing
- 2001

It is proved that there exists a proper vertex colouring f of G such that f( v) ∈ S(v) for each v ∈ V(G) and this proves a weak version of a conjecture of Reed.

### BOREL CHROMATIC NUMBERS

- Mathematics
- 1999

We study in this paper graph coloring problems in the context of descriptive set theory. We consider graphs G=(X, R), where the vertex set X is a standard Borel space (i.e., a complete separable…

### Graph Colouring and the Probabilistic Method

- Computer Science
- 2001

This talk defines graph colouring, explains the probabilistic tools which are used to solve them, and why one would expect the type of tools used to be effective for solving the types of problems typically studied.

### Topics in orbit equivalence

- Physics
- 2004

Preface.- I. Orbit Equivalence.- II. Amenability and Hyperfiniteness.- III. Costs of Equivalence Relations and Groups.- References.- Index.

### Linear-Time and Efficient Distributed Algorithms for List Coloring Graphs on Surfaces

- Mathematics, Computer Science2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
- 2019

The first linear-time algorithms to find fixed parameter tractable with genus as the parameter for 5-coloring n-vertex planar graphs in polylogarithmic rounds are provided.

### Ergodic theorems for the shift action and pointwise versions of the Abért-Weiss theorem

- MathematicsIsrael Journal of Mathematics
- 2019

Let Γ be a countably infinite group. A common theme in ergodic theory is to start with a probability measure-preserving (p.m.p.) action Γ ↷ ( X, μ ) and a map f ∈ L 1 ( X, μ ), and to compare the…

### Unfriendly colorings of graphs with finite average degree

- MathematicsProceedings of the London Mathematical Society
- 2020

In an unfriendly coloring of a graph the color of every node mismatches that of the majority of its neighbors. We show that every probability measure preserving Borel graph with finite average degree…

### Distributed coloring of graphs with an optimal number of colors

- Mathematics, Computer ScienceSTACS
- 2019

This paper studies sufficient conditions to obtain efficient distributed algorithms coloring graphs optimally (i.e. with the minimum number of colors) in the LOCAL model of computation to prove that for infinitely many values of $\Delta, the lower bound $\Delta-k_\Delta+1$ is best possible.

### The list chromatic number of graphs with small clique number

- MathematicsJ. Comb. Theory, Ser. B
- 2019

### Improved Distributed Δ-Coloring

- Computer ScienceDistributed Comput.
- 2021

Two randomized distributed algorithms are presented that improve on an O ( log 3 n / log Δ ) -round algorithm of Panconesi and Srinivasan (STOC'93) and get (exponentially) closer to an Ω ( log log n ) round lower bound of Brandt et al. (ST OC'16).