A Fine-grained Classification of Subquadratic Patterns for Subgraph Listing and Friends
Karl Bringmann, Egor Gorbachev
TL;DR
This work provides a comprehensive fine-grained complexity classification for subgraph problems with a fixed pattern $H$ in colored hosts. It introduces a unifying encoding framework (H-Encoding) and a two-layer decomposition using clique separators, plus a novel hyper-degree splitting technique that significantly extends high/low-degree methods. The main theorem characterises all subquadratic pattern graphs via the decomposition $D(H)$; if any member of $D(H)$ lies outside the simple family $\mathcal{P}$, then Min-Weight-$H$-Subgraph, $H$-Listing, and $H$-Enumeration admit quadratic-time lower bounds under standard hypotheses, otherwise $D(H)$ consists of $P(\alpha,\beta,\gamma\times 2)$-type graphs and the paper provides tight subquadratic upper bounds with an explicit exponent $c(H)=2-1/\mathcal{F}(\alpha_i,\beta_i,\gamma_i)$. The results extend classic triangle and cycle bounds to a broad class of patterns, offering a unified, constructive approach with both conditional lower bounds and matching subquadratic algorithms. The framework also connects to encodings for subgraph finding, tree decompositions, and the study of pattern families in database join theory, highlighting potential practical impact for subgraph search and motif discovery under subquadratic regimes. Key contributions include the definition and exploitation of P-graphs, the Savings function $\mathcal{F}$, the hyper-degree splitting technique, and the comprehensive upper/lower bound machinery for min-weight, listing, and enumeration tasks. Mathematical tools hinge on clique embeddings, induced minors, and reductions guided by well-established fine-grained hypotheses such as 3SUM, MinConv, MinClique, and ZeroClique.
Abstract
In an $m$-edge host graph $G$, all triangles can be listed in time $O(m^{1.5})$ [Itai, Rodeh '78], and all $k$-cycles can be listed in time $O(m^{2-1/{\lceil k/2 \rceil}} + t)$ where $t$ is the output size [Alon, Yuster, Zwick '97]. These classic results also hold for the colored problem variant, where the nodes of the host graph $G$ are colored by nodes in the pattern graph $H$, and we are only interested in subgraphs of $G$ that are isomorphic to the pattern $H$ and respect the colors. We study the problem of listing all $H$-subgraphs in the colored setting, for fixed pattern graphs $H$. As our main result, we determine all pattern graphs $H$ such that all $H$-subgraphs can be listed in subquadratic time $O(m^{2-\varepsilon} + t)$, where $t$ is the output size. Moreover, for each such subquadratic pattern $H$ we determine the smallest exponent $c(H)$ such that all $H$-subgraphs can be listed in time $O(m^{c(H)} + t)$. This is a vast generalization of the classic results on triangles and cycles. To prove this result, we design new listing algorithms and prove conditional lower bounds based on standard hypotheses from fine-grained complexity theory. In our algorithms, we use a new ingredient that we call hyper-degree splitting, where we split tuples of nodes into high degree and low degree depending on their number of common neighbors. We also show the same results for two related problems: finding an $H$-subgraph of minimum total edge-weight in time $O(m^{c(H)})$, and enumerating all $H$-subgraphs in $O(m^{c(H)})$ preprocessing time and constant delay. Again we determine all pattern graphs $H$ that have complexity $c(H) < 2$, and for each such subquadratic pattern we determine the optimal complexity $c(H)$.