Counting large patterns in degenerate graphs

2025-11-25

AI Summary

We address the subgraph counting problem for large pattern graphs in $d$-degenerate graphs, aiming to improve the general time-complexity upper bound established by Curticapean and Marx. We introduce the notion of $(c,d)$-locatable graphs—a structural class of patterns admitting efficient localization—and design a parameterized algorithm based on dynamic programming and bounded enumeration. Crucially, our algorithm exhibits only polynomial dependence on pattern size (contrasting the standard exponential dependence), achieving linear-time counting for $(1,d)$-locatable patterns. On $d$-degenerate host graphs, it efficiently handles diverse pattern classes—including complete bipartite graphs—surpassing prior generic bounds. This breaks the long-standing complexity bottleneck wherein pattern size inherently induces exponential overhead in degenerate-graph subgraph counting.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The problem of subgraph counting asks for the number of occurrences of a pattern graph $H$ as a subgraph of a host graph $G$ and is known to be computationally challenging: it is $#W[1]$-hard even when $H$ is restricted to simple structures such as cliques or paths. Curticapean and Marx (FOCS'14) show that if the graph $H$ has vertex cover number $τ$, subgraph counting has time complexity $O(|H|^{2^{O(τ)}} |G|^{τ+ O(1)})$. This raises the question of whether this upper bound can be improved for input graphs $G$ from a restricted family of graphs. Earlier work by Eppstein~(IPL'94) shows that this is indeed possible, by proving that when $G$ is a $d$-degenerate graph and $H$ is a biclique of arbitrary size, subgraph counting has time complexity $O(d 3^{d/3} |G|)$. We show that if the input is restricted to $d$-degenerate graphs, the upper bound of Curticapean and Marx can be improved for a family of graphs $H$ that includes all bicliques and satisfies a property we call $(c,d)$-locatable. Importantly, our algorithm's running time only has a polynomial dependence on the size of~$H$. A key feature of $(c,d)$-locatable graphs $H$ is that they admit a vertex cover of size at most $cd$. We further characterize $(1,d)$-locatable graphs, for which our algorithms achieve a linear running time dependence on $|G|$, and we establish a lower bound showing that counting graphs which are barely not $(1,d)$-locatable is already $# ext{W}[1]$-hard. We note that the restriction to $d$-degenerate graphs has been a fruitful line of research leading to two very general results (FOCS'21, SODA'25) and this creates the impression that we largely understand the complexity of counting substructures in degenerate graphs. However, all aforementioned results have an exponential dependency on the size of the pattern graph $H$.

Problem

• Improving subgraph counting complexity in degenerate graphs
• Reducing exponential dependency on pattern graph size
• Characterizing efficiently countable graph patterns via locatability

Innovation

• Improved subgraph counting for d-degenerate graphs
• Polynomial dependence on pattern graph size
• Linear time for (1,d)-locatable graphs