# Semantic Width of Conjunctive Queries and Constraint Satisfaction Problems

## TL;DR

This paper introduces a unified concept called semantic width that connects various structural and semantic approaches to analyzing the complexity of conjunctive queries and constraint satisfaction problems, aiming to identify tractable cases.

## Contribution

It proposes a general notion of semantic width that unifies existing structural and semantic methods for analyzing CQs and CSPs, extending the understanding of their complexity.

## Key findings

- Introduces semantic versions of fractional hypertree width, adaptive width, submodular width, and fractional cover number.
- Connects all three traditional research threads in CQ and CSP complexity analysis.
- Provides a framework for identifying tractable subclasses based on semantic width.

## Abstract

Answering Conjunctive Queries (CQs) and solving Constraint Satisfaction Problems (CSPs) are arguably among the most fundamental tasks in Computer Science. They are classical NP-complete problems. Consequently, the search for tractable fragments of these problems has received a lot of research interest over the decades. This research has traditionally progressed along three orthogonal threads. a) Reformulating queries into simpler, equivalent, queries (semantic optimization) b) Bounding answer sizes based on structural properties of the query c) Decomposing the query in such a way that global consistency follows from local consistency. Much progress has been made by various works that connect two of these threads. Bounded answer sizes and decompositions have been shown to be tightly connected through the important notions of fractional hypertree width and, more recently, submodular width. recent papers by Barcel\'o et al. study decompositions up to generalized hypertree width under semantic optimization. In this work, we connect all three of these threads by introducing a general notion of semantic width and investigating semantic versions of fractional hypertree width, adaptive width, submodular width and the fractional cover number.