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

TL;DR

This paper classifies local problems on oriented paths based on their solvability with Borel, measurable, iid, or Baire solutions, revealing that randomness does not aid in solving these problems and unifying multiple fields' classifications.

Contribution

It provides a comprehensive classification of local problems on paths across descriptive combinatorics, distributed computing, and probability, showing the equivalence of solution classes and the non-utility of randomness.

Findings

Four classes of local problems identified

Most classes have definitions across all three fields

Randomness does not improve problem-solving on paths

Abstract

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 processes [Holroyd et al. Annals of Prob. 2017, Greb\'ik, Rozho\v{n} arXiv:2103.08394], this finishes the classification of local problems with inputs on oriented paths using complexity classes from these three fields. A simple picture emerges: there are four classes of local problems and most classes have natural definitions in all three fields. Moreover, we now know that randomness does \emph{not} help with solving local problems on oriented paths.