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

@article{Grebik2021ClassificationOL, title={Classification of Local Problems on Paths from the Perspective of Descriptive Combinatorics}, author={Jan Greb'ik and V{\'a}clav Rozhoň}, journal={ArXiv}, year={2021}, volume={abs/2103.14112} }

We classify which local problems with inputs on oriented paths have so-called Borel solution and show that this class of problems remains the same if we instead require a measurable solution, a factor of iid solution, or a solution with the property of Baire. Together with the work from the field of distributed computing [Balliu et al. PODC 2019], the work from the field of descriptive combinatorics [Gao et al. arXiv:1803.03872, Bernshteyn arXiv:2004.04905] and the work from the field of random…

## 6 Citations

### 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.

### 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.

### On Homomorphism Graphs

- MathematicsArXiv
- 2021

It is shown that for ∆ > 2 it is impossible to give a simple characterization of acyclic ∆-regular Borel graphs with Borel chromatic number at most ∆: such graphs form a Σ12-complete set, implying a strong failure of Brooks’-like theorems in the Borel context.

### Properly learning monotone functions via local reconstruction

- Computer ScienceArXiv
- 2022

—We give a 2 ˜ O ( √ n/ε ) -time algorithm for properly learning monotone Boolean functions under the uni- form distribution over { 0 , 1 } n . Our algorithm is robust to adversarial label noise and…

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