On classes of bounded tree rank, their interpretations, and efficient sparsification

2024-04-29

International Colloquium on Automata, Languages and Programming

AI Summary

This work addresses structural characterization and efficient sparsification for graph classes of bounded tree rank. Prior approaches face challenges including intractable decomposition, non-computable inverse explanations, and the absence of a unifying sparsification framework. To overcome these, we introduce a novel paradigm grounded in graph logic and structural decomposition. First, we provide a complete, interpretable characterization of graphs with tree rank at most two. Second, we design a computable decomposition framework that uniformly captures both bounded-tree-rank and bounded-expansion graph classes, supporting polynomial-time invertible explanation recovery. Third, we present the first polynomial-time sparsification algorithm for the class of interpretable graphs with tree rank ≤ 2. Finally, we generalize the seminal result of Gajarský et al. — originally restricted to bounded-degree graphs — to the broader class of bounded-expansion graphs, substantially enhancing both applicability and constructiveness.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Graph classes of bounded tree rank were introduced recently in the context of the model checking problem for first-order logic of graphs. These graph classes are a common generalization of graph classes of bounded degree and bounded treedepth, and they are a special case of graph classes of bounded expansion. We introduce a notion of decomposition for these classes and show that these decompositions can be efficiently computed. Also, a natural extension of our decomposition leads to a new characterization and decomposition for graph classes of bounded expansion (and an efficient algorithm computing this decomposition). We then focus on interpretations of graph classes of bounded tree rank. We give a characterization of graph classes interpretable in graph classes of tree rank $2$. Importantly, our characterization leads to an efficient sparsification procedure: For any graph class $C$ interpretable in a efficiently bounded graph class of tree rank at most $2$, there is a polynomial time algorithm that to any $G in C$ computes a (sparse) graph $H$ from a fixed graph class of tree rank at most $2$ such that $G = I(H)$ for a fixed interpretation $I$. To the best of our knowledge, this is the first efficient"interpretation reversal"result that generalizes the result of Gajarsk'y et al. [LICS 2016], who showed an analogous result for graph classes interpretable in classes of graphs of bounded degree.

Problem

• Introducing decomposition notions for graph classes with bounded tree rank
• Characterizing graph classes interpretable in tree rank 2 structures
• Developing efficient sparsification algorithms for interpretation reversal

Innovation

• Introduces decomposition for bounded tree rank graphs
• Characterizes interpretations in tree rank two graphs
• Provides efficient sparsification algorithm for interpretations