On Injective Embeddings of Tree Patterns

TL;DR

This paper analyzes different types of injective embeddings of tree patterns, exploring their hierarchical relationships and computational complexities, including how restrictions on patterns affect model checking difficulty.

Contribution

It provides a comprehensive complexity analysis of various injective tree pattern embeddings and examines how pattern restrictions influence computational hardness.

Findings

Hierarchy of embedding types with varying complexities

Model checking is P or NP-complete depending on the embedding type

Restrictions on pattern structure can reduce complexity

Abstract

We study three different kinds of embeddings of tree patterns: weakly-injective, ancestor-preserving, and lca-preserving. While each of them is often referred to as injective embedding, they form a proper hierarchy and their computational properties vary (from P to NP-complete). We present a thorough study of the complexity of the model checking problem i.e., is there an embedding of a given tree pattern in a given tree, and we investigate the impact of various restrictions imposed on the tree pattern: bound on the degree of a node, bound on the height, and type of allowed labels and edges.