# Point-width and Max-CSPs

## TL;DR

This paper introduces a new hypergraph decomposition framework that unifies existing tractability conditions for Max-CSPs, providing a broader understanding of structural restrictions that ensure computational feasibility.

## Contribution

It proposes point decompositions as a new hypergraph framework, generalizing bounded MIM-width and β-acyclicity, and offers a new characterization of MIM-width.

## Key findings

- Introduces point decompositions for hypergraphs.
- Provides a new sufficient condition for Max-CSP tractability.
- Characterizes bounded MIM-width and discusses related properties.

## Abstract

The complexity of (unbounded-arity) Max-CSPs under structural restrictions is poorly understood. The two most general hypergraph properties known to ensure tractability of Max-CSPs, $\beta$-acyclicity and bounded (incidence) MIM-width, are incomparable and lead to very different algorithms. We introduce the framework of point decompositions for hypergraphs and use it to derive a new sufficient condition for the tractability of (structurally restricted) Max-CSPs, which generalises both bounded MIM-width and \b{eta}-acyclicity. On the way, we give a new characterisation of bounded MIM-width and discuss other hypergraph properties which are relevant to the complexity of Max-CSPs, such as $\beta$-hypertreewidth.