Space Complexity Dichotomies for Subgraph Finding Problems in the Streaming Model

TL;DR

This work provides a comprehensive set of dichotomy theorems for four fixed-pattern subgraph problems in insertion-only streaming: Sub(H) and IndSub(H) on undirected graphs, and Sub(\vec H) and IndSub(\vec H) on oriented graphs. Central to the results are Turán-type bounds ex$(n,H)$ and the well-/non-well-oriented (WO/NWO) structure of patterns, which determine whether near-quadratic space is necessary or whether efficient $\tilde O(1)$-pass or even single-pass algorithms exist. Most undirected and oriented induced-pattern problems are hard (requiring $\tilde\Omega(n^2/p)$ space in $p$ passes) except for a small set of patterns ($P_3$, $P_4$, co-$P_3$ in the undirected induced case, and co-$P_3$ in the induced oriented case) where clever sparse certificates or color-coding enable space-efficient algorithms. In particular, Sub$(H)$ undergoes a sharp bipartite/non-bipartite dichotomy mirroring Turán-type behavior, while the induced variants exhibit a small exceptional set of tractable patterns and otherwise near-quadratic lower bounds, with the oriented versions showing nuanced multi-pass behaviors driven by WO/NWO structure. The results have implications for streaming graph algorithms, highlighting when ultra-efficient space is possible and when the complexity is forced by underlying extremal properties and orientation structure.

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 $n^{2-Ω(1)}$-space algorithm if and only if $H$ is bipartite. (2) IndSub$(H)$ can be solved by an $\tilde{O}(1)$-pass $n^{2-Ω(1)}$-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 $n^{2-Ω(1)}$-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 $n^{2-Ω(1)}$-space algorithm if and only if the underlying undirected simple graph $H$ is a $co\mbox{-}P_3$.

Figures (2)

• Figure 3: The structure of an induced-$P_4$-free graph. An edge between two groups means that all vertices in one are adjacent to all vertices in the other.
• Figure 7: An example where $H\subseteq G_H$ but the parties have no common element.

Theorems & Definitions (57)

• Definition 2: Insertion-Only Graph Streaming Model McGregor14
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Definition 9: well-oriented
• Definition 10: Turán number
• Theorem : Erdős-Stone EST46