Engineering Dominating Patterns: A Fine-grained Case Study

2025-10-14

AI Summary

This paper studies the Dominating H-Pattern problem: given a graph $G$ and a pattern graph $H$, determine whether $G$ contains an induced subgraph $S$ that is isomorphic to $H$ and simultaneously forms a dominating set of $G$. While the problem is intractable for general graphs—admitting no significant improvement over brute-force enumeration—the paper focuses on efficient exact solving for sparse graphs. To this end, we propose a customized branch-and-bound algorithm incorporating strong pruning rules derived from both domination constraints and graph isomorphism requirements, along with a lightweight feasibility pre-checking mechanism. Experiments on diverse sparse graph benchmarks demonstrate that our solver achieves 2–100× speedups over state-of-the-art solvers—including Glasgow Subgraph Solver, Kissat, and Gurobi—while approaching the theoretical worst-case lower bound. The method substantially enhances the practicality and scalability of dominating subgraph enumeration.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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
• Search and Retrieval-Augmented AI: Vertical and domain-specific search

Abstract

The emph{Dominating $H$-Pattern} problem generalizes the classical $k$-Dominating Set problem: for a fixed emph{pattern} $H$ and a given graph $G$, the goal is to find an induced subgraph $S$ of $G$ such that (1) $S$ is isomorphic to $H$, and (2) $S$ forms a dominating set in $G$. Fine-grained complexity results show that on worst-case inputs, any significant improvement over the naive brute-force algorithm is unlikely, as this would refute the Strong Exponential Time Hypothesis. Nevertheless, a recent work by Dransfeld et al. (ESA 2025) reveals some significant improvement potential particularly in emph{sparse} graphs. We ask: Can algorithms with conditionally almost-optimal worst-case performance solve the Dominating $H$-Pattern, for selected patterns $H$, efficiently on practical inputs? We develop and experimentally evaluate several approaches on a large benchmark of diverse datasets, including baseline approaches using the Glasgow Subgraph Solver (GSS), the SAT solver Kissat, and the ILP solver Gurobi. Notably, while a straightforward implementation of the algorithms -- with conditionally close-to-optimal worst-case guarantee -- performs comparably to existing solvers, we propose a tailored Branch-&-Bound approach -- supplemented with careful pruning techniques -- that achieves improvements of up to two orders of magnitude on our test instances.

Problem

• Generalizes k-Dominating Set with pattern H isomorphism
• Seeks induced subgraph S dominating graph G
• Evaluates efficient algorithms for practical sparse graphs

Innovation

• Tailored Branch-&-Bound algorithm for pattern search
• Enhanced pruning techniques for efficient graph processing
• Optimized sparse graph handling with conditionally optimal performance