Space Complexity Dichotomies for Subgraph Finding Problems in the Streaming Model

TL;DR

This paper establishes a complete classification of the space complexity for various subgraph detection problems in streaming models, revealing precise conditions under which efficient algorithms are possible.

Contribution

It provides a comprehensive dichotomy theorem for four variants of subgraph finding problems in streaming, characterizing when they admit sublinear space algorithms.

Findings

Sub$(H)$ solvable with polylog passes iff H is bipartite.

IndSub$(H)$ solvable iff H is P3, P4, or co-P3.

Sub$(oldsymbol{H})$ solvable iff each component is well-oriented bipartite or a specific tree.

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 $co\mbox{-}P_3$.