Space Complexity Dichotomies for Subgraph Finding Problems in the Streaming Model

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 $co\mbox{-}P_3$.

💡 Research Summary

The paper investigates the space complexity of four variants of the subgraph‑finding problem in the insertion‑only streaming model: Sub(H), IndSub(H), Sub(→H), and IndSub(→H), where H is a fixed undirected pattern and →H is a fixed oriented pattern. The authors establish complete dichotomy theorems that separate pattern graphs into two classes: those that admit an Õ(1)‑pass algorithm using n^{2‑Ω(1)} space, and those that require essentially Ω(n²) space even with many passes.

The analysis starts from the classical Turán number ex(n,H), which measures the maximum number of edges in an n‑vertex H‑free graph. By the Erdős–Stone theorem, ex(n,H)=o(n²) iff H is bipartite. The authors extend the Alon‑Krivelich‑Sudakov bound to show ex(n,H)=O(n^{2‑1/Δ′(H)}) for any bipartite H, where Δ′(H) is the smaller of the maximum degrees on the two sides of a bipartition. This yields a simple “naïve” streaming algorithm: keep any subgraph with more than ex(n,H) edges; it must contain H. Consequently, Sub(H) can be solved with Õ(1) passes and n^{2‑Ω(1)} space exactly when H is bipartite (Theorem 3). For non‑bipartite H, a reduction from the multi‑party set‑disjointness communication problem shows that any p‑pass algorithm needs Ω(n²/p) space, establishing the lower bound.

For the induced version IndSub(H), the situation is dramatically harder because a dense graph can avoid many induced subgraphs. The authors prove that, except for three tiny patterns—P₃ (a path on three vertices), P₄ (a path on four vertices), and co‑P₃ (the complement of P₃)—any Õ(1)‑pass algorithm must use Ω(n²) space (Theorem 5). For these three exceptional graphs, they design Õ(1)‑pass, Õ(n)‑space algorithms based on a structural decomposition of cographs and a novel “small‑forest‑preserving certificate” that stores only O(|H|) cores, each of size O(|H|), sufficient to certify the existence of the required induced subgraph.

The oriented case introduces the notions of well‑oriented (WO) and non‑well‑oriented (NWO) graphs. A WO oriented bipartite graph admits a bipartition such that every arc points from one side to the other; an NWO vertex has both indegree and outdegree non‑zero. Extending the Turán‑type bound to oriented graphs, the authors show that for WO bipartite →H, ex(n,→H)=O(n^{2‑1/Δ′(H)}) (Corollary 11). Hence Sub(→H) is easy (single‑pass, n^{2‑Ω(1)} space) exactly for WO components. For NWO components, the picture is subtler. If each connected component of →H is either WO or a tree containing at most one NWO vertex, then a single‑pass algorithm using Õ(n) space exists (Theorem 7). This algorithm builds a sparse certificate consisting of a collection of small cores that preserve any forest rooted at designated vertices, allowing the detection of the oriented pattern without storing the whole graph. Conversely, if a component contains two NWO vertices (NWOC₂) or a NWO vertex together with a cycle (NWOC₁+ C), the authors prove Ω(n²) lower bounds via reductions from multi‑disjointness (Theorem 6). Thus Sub(→H) admits an efficient algorithm iff every component satisfies the WO‑or‑single‑NWO‑tree condition.

The induced oriented problem IndSub(→H) mirrors the undirected induced case. Using reductions from set‑disjointness, the authors show that all patterns are hard except when the underlying undirected graph is co‑P₃. In that case, an Õ(1)‑pass, Õ(n)‑space algorithm exists (Theorem 8).

The paper’s technical contributions include:

1. Tight connections between Turán numbers and streaming space for both undirected and oriented graphs.
2. Novel communication‑complexity lower bounds for a wide range of patterns, employing multi‑party disjointness and set‑disjointness.
3. The introduction of “sparse certificates” and “small‑forest‑preserving certificates” that enable sublinear‑space detection of certain oriented trees with limited NWO vertices.
4. A complete classification (dichotomy) for each of the four problem variants, summarised in Table 1.

The work closes a gap in the literature by showing that, contrary to intuition, many oriented patterns with Θ(n²) Turán numbers are still easy in the streaming model due to structural restrictions on NWO vertices. It also highlights the delicate interplay between graph orientation, bipartiteness, and induced versus non‑induced subgraph detection, providing a comprehensive framework for future research on streaming graph algorithms and their inherent space limits.