Space Complexity Dichotomies for Subgraph Finding Problems in the Streaming Model

2026-02-08

AI Summary

This work investigates the space complexity of four fundamental subgraph finding problems in the streaming model—namely, detecting subgraphs and induced subgraphs in both undirected and directed graphs. By integrating structural graph properties such as bipartiteness and well-orientability with complexity-theoretic reductions, the paper establishes the first tight dichotomy theorems for the space complexity of these four problems. The central contribution is a precise characterization of the necessary and sufficient conditions under which these problems admit efficient solutions using near-constant passes and subquadratic space. For instance, it shows that undirected subgraph detection is solvable in subquadratic space if and only if the pattern graph is bipartite, thereby clearly delineating the boundary of pattern graphs that permit efficient streaming algorithms.

Technology Category

• Data Mining & Knowledge Management: Data Stream Mining
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We study the space complexity of four variants of the standard subgraph finding problem in the streaming model. Specifically, given an $n$-vertex input graph and a fixed-size pattern graph, we consider two settings: undirected simple graphs, denoted by $G$ and $H$, and oriented graphs, denoted by $\vec{G}$ and $\vec{H}$. Depending on the setting, the task is to decide whether $G$ contains $H$ as a subgraph or as an induced subgraph, or whether $\vec{G}$ contains $\vec{H}$ as a subgraph or as an induced subgraph. Let Sub$(H)$, IndSub$(H)$, Sub$(\vec{H})$, and IndSub$(\vec{H})$ denote these four variants, respectively. An oriented graph is well-oriented if it admits a bipartition in which every arc is oriented from one part to the other, and a vertex is non-well-oriented if both its in-degree and out-degree are non-zero. For each variant, we obtain a complete dichotomy theorem, briefly summarized as follows. (1) Sub$(H)$ can be solved by an $\tilde{O}(1)$-pass $n^{2-\Omega(1)}$-space algorithm if and only if $H$ is bipartite. (2) IndSub$(H)$ can be solved by an $\tilde{O}(1)$-pass $n^{2-\Omega(1)}$-space algorithm if and only if $H \in \{P_3, P_4, co\mbox{-}P_3\}$. (3) Sub$(\vec{H})$ can be solved by a single-pass $n^{2-\Omega(1)}$-space algorithm if and only if every connected component of $\vec H$ is either a well-oriented bipartite graph or a tree containing at most one non-well-oriented vertex. (4) IndSub$(\vec{H})$ can be solved by an $\tilde{O}(1)$-pass $n^{2-\Omega(1)}$-space algorithm if and only if the underlying undirected simple graph $H$ is a $co\mbox{-}P_3$.

Problem

• subgraph finding
• streaming model
• space complexity
• dichotomy
• induced subgraph

Innovation

• streaming algorithms
• subgraph finding
• space complexity
• dichotomy theorem
• well-oriented graphs