Subgraph Counting in Subquadratic Time for Bounded Degeneracy Graphs

TL;DR

This work advances exact subgraph counting on bounded-degeneracy graphs by introducing a unifying reduction framework that maps homomorphism counting for directed patterns to counting weighted, colorful copies of smaller patterns in reduced graphs. For all patterns H with at most 9 vertices, Sub_H(G) can be computed in time $f(\kappa)\tilde{O}(n^{5/3})$, where $\kappa$ is the degeneracy of $G$, combining orientation-based techniques with hypergraph counting. It also delivers subquadratic cycle counting up to length 10, guided by the cycle-counting exponents $d_k$ from the best known algorithms, and establishes conditional hardness results via reductions to hypergraph counting and subdivisions. The core contribution is a flexible framework that reduces a broad class of subgraph counting problems to counting smaller hypergraphs in general graphs, enabling a path toward a general theory of subgraph counting in bounded-degeneracy graphs with potential practical impact on large real-world networks.

Abstract

We study the classic problem of subgraph counting, where we wish to determine the number of occurrences of a fixed pattern graph $H$ in an input graph $G$ of $n$ vertices. Our focus is on bounded degeneracy inputs, a rich family of graph classes that also characterizes real-world massive networks. Building on the seminal techniques introduced by Chiba-Nishizeki (SICOMP 1985), a recent line of work has built subgraph counting algorithms for bounded degeneracy graphs. Assuming fine-grained complexity conjectures, there is a complete characterization of patterns $H$ for which linear time subgraph counting is possible. For every $r \geq 6$, there exists an $H$ with $r$ vertices that cannot be counted in linear time. In this paper, we initiate a study of subquadratic algorithms for subgraph counting on bounded degeneracy graphs. We prove that when $H$ has at most $9$ vertices, subgraph counting can be done in $\tilde{O}(n^{5/3})$ time. As a secondary result, we give improved algorithms for counting cycles of length at most $10$. Previously, no subquadratic algorithms were known for the above problems on bounded degeneracy graphs. Our main conceptual contribution is a framework that reduces subgraph counting in bounded degeneracy graphs to counting smaller hypergraphs in arbitrary graphs. We believe that our results will help build a general theory of subgraph counting for bounded degeneracy graphs.

Figures (15)

• Figure 1: (a) The $6$-cycle obstruction, this orientation has three sources intersecting with each other, this oriented pattern can not be counted in linear time in bounded degeneracy graphs. Adding an edge connecting the end-points of every out-out wedge gives a triangle. (b) An example of how the oriented ${\cal C}_{8}$ reduces to a ${\cal C}_{4}$, the four sources become edges connecting the intersection vertices.
• Figure 2: Two more complex examples $P$-reducibility.
• Figure 3: An example of the construction of $G_{{\cal C}_{3}}$, for pattern $\vec{H}$ and input graph $\vec{G}$. The red vertices correspond with $i_1$ in $\vec{H}$, the green ones with $i_2$ and the blue ones with $i_3$. The weight of the edges is $1$ except when indicated. For example there are two homomorphisms $\phi: \vec{H}(s_2) \to \vec{G}$ that map $i_1$ and $i_2$ to $c$, hence the edge $(c- 1,c- 2)$ has weight $2$. One can verify that the number of homomorphisms from $\vec{H}$ to $\vec{G}$ is equal to the sum of products of (colorful) triangles in $G_{{\cal C}_{3}}$.
• Figure 4: The hypergraph ${\cal H}_\triangle$.
• Figure 5: The diamond graph $\mathcal{D}$, the hypergraph ${\cal H}_1$ and the hypergraph ${\cal H}_2$.

Theorems & Definitions (72)

• Theorem 1
• Theorem 2
• Lemma 2
• Lemma 2
• Lemma 2
• Lemma 2
• Corollary 2
• Conjecture 3
• Lemma 3
• Definition 4: DAG-tree decomposition Br19