Distributed Model Checking in Graphs Classes of Bounded Expansion

2024-11-22

arXiv.org

AI Summary

This paper presents the first distributed first-order logic (FO) model-checking algorithm for bounded-expansion graph classes (e.g., planar graphs, graphs of bounded treewidth) in the CONGEST model, achieving optimal round complexity O(D + log n). The algorithm uniformly supports FO formula evaluation, optimization, counting, and locally verifiable certification. Methodologically, it integrates low-density graph decomposition, local structural feature extraction, hierarchical centralized message aggregation, and a syntax-tree-driven distributed evaluation protocol. Unlike prior work restricted to graphs of bounded tree depth, our approach extends applicability to the full class of bounded-expansion graphs. The round complexity is asymptotically optimal—tight up to a logarithmic factor—and the certification size is only O(log n) bits, markedly improving upon the Ω(√n) lower bound for general graphs.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic 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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We show that for every first-order logic (FO) formula $varphi$, and every graph class $mathcal{G}$ of bounded expansion, there exists a distributed (deterministic) algorithm that, for every $n$-node graph $Ginmathcal{G}$ of diameter $D$, decides whether $Gmodels varphi$ in $O(D+log n)$ rounds under the standard CONGEST model. Graphs of bounded expansion encompass many classes of sparse graphs such as planar graphs, bounded-treedepth graphs, bounded-treewidth graphs, bounded-degree graphs, and graphs excluding a fixed graph $H$ as a minor or topological minor. Note that our algorithm is optimal up to a logarithmic additional term, as even a simple FO formula such as"there are two vertices of degree 3"already on trees requires $Omega(D)$ rounds in CONGEST. Our result extends to solving optimization problems expressed in FO (e.g., $k$-vertex cover of minimum weight), as well as to counting the number of solutions of a problem expressible in a fragment of FO (e.g., counting triangles), still running in $O(D+log n)$ rounds under the CONGEST model. This exemplifies the contrast between sparse graphs and general graphs as far as CONGEST algorithms are concerned. For instance, Drucker, Kuhn, and Oshman [PODC 2014] showed that the problem of deciding whether a general graph contains a 4-cycle requires $Theta(sqrt{n}/log n)$ rounds in CONGEST. For counting triangles, the best known algorithm of Chang, Pettie, and Zhang [SODA 2019] takes $ ilde{O}(sqrt{n})$ rounds. Finally, our result extends to distributed certification. We show that, for every FO formula~$varphi$, and every graph class of bounded expansion, there exists a certification scheme for $varphi$ using certificates on $O(log n)$ bits. This significantly generalizes the recent result of Feuilloley, Bousquet, and Pierron [PODC 2022], which held solely for graphs of bounded treedepth.

Problem

• Distributed FO model checking on bounded expansion graphs
• Solving local FO formulas in O(log n) rounds
• Deciding FO formulas in O(D + log n) rounds

Innovation

• Distributed FO model checking on bounded expansion graphs
• Deterministic algorithm for local FO formulas in O(log n) rounds
• Deterministic FO decision in O(D + log n) rounds