insight - Algorithms and Data Structures - # Parallel Canonization of Graphs with Bounded Rank-Width

Core Concepts

Graphs of bounded rank-width can be canonized efficiently in parallel using a TC2 circuit, by leveraging the Weisfeiler-Leman algorithm to identify and canonize the graph components.

Abstract

The paper presents an efficient parallel algorithm for computing canonical labelings of graphs with bounded rank-width. The key insights are:
The (6k+3)-dimensional Weisfeiler-Leman (WL) algorithm can identify graphs of rank-width k in O(log n) rounds, allowing for a TC1 isomorphism test.
By tracking the depth of the recursion and leveraging the balanced rank decomposition of graphs of bounded rank-width, the authors construct a TC2 circuit for computing canonical labelings.
The main steps are:
Use the framework of Grohe and Neuen to descend along a rank decomposition, producing a canonical labeling.
Ensure the choices are canonical by utilizing the Weisfeiler-Leman algorithm.
Establish that the (6k+3)-dimensional WL algorithm identifies graphs of rank-width k in O(log n) rounds, enabling a TC1 isomorphism test.
Combine the parallel WL implementation with the rank decomposition to obtain a TC2 bound for canonization.
The authors also show that graphs of bounded treewidth can be identified by the (3k+6)-dimensional WL algorithm in O(log n) rounds, improving the previous descriptive complexity results.

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by Michael Leve... at **arxiv.org** 04-26-2024

Deeper Inquiries

The rank decomposition of graphs with bounded rank-width can be computed efficiently in parallel. The rank decomposition is a hierarchical structure that splits the vertices of a graph into subsets based on cuts of low complexity. This decomposition can be utilized to efficiently compute canonical labelings of graphs of bounded rank-width. The framework of split pairs and flip functions, as introduced by Grohe & Neuen, allows for the efficient computation of rank decompositions. By leveraging these split pairs and flip functions, it is possible to control the depth of the recursion in the computation, leading to an efficient parallel algorithm for computing the rank decomposition of graphs with bounded rank-width.

Beyond rank-width and treewidth, there are other structural properties of graphs that can be leveraged to obtain efficient parallel isomorphism and canonization algorithms. For example, properties such as tree-depth, path-width, and clique-width can also provide insights into the structure of graphs and aid in developing efficient algorithms for isomorphism testing and canonization. These properties define different ways in which graphs can be decomposed or represented, and algorithms can be designed to exploit these structural characteristics to achieve computational efficiency. By exploring a diverse range of structural properties, researchers can uncover new avenues for developing efficient algorithms for graph analysis tasks.

The improved descriptive complexity results for graphs of bounded treewidth have significant implications in terms of their logical characterization and potential applications. The descriptive complexity results provide insights into the expressive power of logical formulas in characterizing graphs of bounded treewidth. By showing that these graphs can be identified using formulas with specific quantifier depths and variable counts, the results offer a deeper understanding of the complexity of properties expressible in logic for this graph class. This has implications for model-checking, database query evaluation, and other applications where logical formulas are used to represent and reason about graph structures. The improved descriptive complexity results open up new possibilities for efficiently handling graphs of bounded treewidth in various computational tasks.

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