Fine-Grained Complexity of Subgraph Isomorphism: Characterizing Patterns with Subquadratic Algorithms

Core Concepts

For any pattern graph H, we determine whether it admits subquadratic algorithms for minimum-weight subgraph, subgraph listing, and subgraph enumeration, and if so, we provide the optimal time complexity.

Abstract

The paper studies the subgraph isomorphism problem, where the pattern graph H is fixed and the task is to find or list all occurrences of H in a given host graph G. The authors focus on the setting where the pattern H is small and the goal is to obtain algorithms with time complexity that is subquadratic in the number of edges m of the host graph.
The key insights are:

The authors introduce the concept of "P-graphs", a class of patterns that can be constructed by stitching together certain basic building blocks along cliques. They show that all patterns with subquadratic complexity can be decomposed into P-graphs.

For each P-graph, the authors define a "savings function" F(α,β,γ) that captures the time complexity of algorithms for that pattern. The overall complexity is then determined by the maximum value of 2-1/F(α,β,γ) over all the P-graph components.

The authors provide fine-grained conditional lower bounds, showing that patterns not decomposable into P-graphs have complexity at least quadratic. For patterns that are decomposable into P-graphs, they design algorithms matching the lower bounds.

The results provide a complete characterization of pattern graphs that admit subquadratic algorithms for minimum-weight subgraph, subgraph listing, and subgraph enumeration, and determine the optimal time complexity for each such pattern.

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Key Insights Distilled From

A Fine-grained Classification of Subquadratic Patterns for Subgraph Listing and Friends

by Karl Bringma... at arxiv.org 04-09-2024

https://arxiv.org/pdf/2404.04369.pdf

Deeper Inquiries

Can the techniques developed in this paper be extended to characterize patterns with complexity better than 2-1/k for some integer k > 2

The techniques developed in this paper can potentially be extended to characterize patterns with complexity better than 2-1/k for some integer k > 2. The current framework focuses on patterns with complexity below 2, but with further research and analysis, it may be possible to adapt the methodology to explore patterns with higher complexities. By refining the approach and possibly introducing new tools or hypotheses, it could be feasible to delve into the realm of patterns with more intricate time complexities.

Are there any connections between the P-graphs identified in this paper and other well-studied graph classes in the literature

The P-graphs identified in this paper exhibit interesting connections to other well-studied graph classes in the literature. For instance, the P-graphs can be linked to series-parallel graphs, as mentioned in the context. Additionally, the structure and characteristics of P-graphs may have similarities or relationships with other graph families such as trees, cycles, and bipartite graphs. Exploring these connections further could provide insights into the broader landscape of graph theory and complexity analysis.

What are the practical implications of this fine-grained complexity analysis for subgraph isomorphism problems in real-world applications

The fine-grained complexity analysis conducted in this paper has significant practical implications for subgraph isomorphism problems in real-world applications. By understanding the time complexities of different patterns and subgraph variations, researchers and practitioners can make informed decisions when designing algorithms for tasks like graph pattern detection, network analysis, and database queries. The results of this analysis can guide the development of efficient algorithms tailored to specific pattern structures, ultimately improving the performance and scalability of subgraph isomorphism solutions in practical scenarios.