Current Challenges in Infinite-Domain Constraint Satisfaction: Dilemmas of the Infinite Sheep

TL;DR

This paper reviews the current state of research on infinite-domain constraint satisfaction problems, highlighting key dilemmas and exploring potential solutions to understand their computational complexity.

Contribution

It provides an overview of mathematical ideas, dilemmas, and future directions in studying CSPs over finitely bounded homogeneous structures.

Findings

Identifies three main dilemmas in infinite-domain CSP research

Explains mathematical frameworks used in the field

Suggests possible approaches to resolve existing challenges

Abstract

A Constraint Satisfaction Problem (CSP) is a computational problem where we are given variables and constraints about them; the question is whether the variables can be assigned values such that all constraints are satisfied. We give an overview of the current state of research on CSPs where values for the variables and constraints are taken from a finitely bounded homogeneous structure which is fixed beforehand. We explain the main mathematical ideas so far, the three dilemmas they brought upon us, and what could be done to overcome them in order to obtain a satisfactory understanding of the computational complexity of such CSPs.