Engineering Dominating Patterns: A Fine-grained Case Study

TL;DR

This paper studies Dominating $H$-Pattern, a generalization of the classical $k$-Dominating Set, focusing on sparse graphs and conditional lower bounds to bridge theory with practice. It presents fast, integrated Branch-&-Bound solvers that jointly leverage domination and subgraph isomorphism to identify dominating $H$-patterns for cycles $C_k$, paths $P_k$, and induced matchings $M_k$, achieving substantial speedups over baselines. The authors combine heuristics (e.g., low-degree-dominator starts and 3-scattered set bounds), pattern enumeration baselines, and specialized solvers for different pattern families, with extensive experiments on real-world and synthetic datasets including OV-based hard instances. The work demonstrates practical, scalable performance improvements and provides a framework that can be extended to more pattern classes, contributing to both structural graph theory and practical graph mining. Overall, the study advances how conditional lower-bound insights can be married with tailored algorithm engineering to solve complex domination-pattern problems in realistic inputs. All code and data are made available for reproducibility, supporting further research and application-aware development in graph-pattern mining.

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.

Figures (8)

• Figure 2.1: Sketch for possible ways to extend a partial assignment for $k$-matching pattern. The black squares are vertices of the host graph that are assigned to some pattern vertex. The other marks show how this partial assignment can be extended properly (green dots) or incorrectly (red crosses). Among all these vertices all those edges (of the host graph) are shown that are incident to some assigned vertex.
• Figure 2.2: Sketch for possible ways to extend a partial assignment for $k$-cycle pattern. The black squares are vertices of the host graph that are assigned to some pattern vertex. The other marks show how this partial assignment can be extended properly (green dots) or incorrectly (red crosses). Among all these vertices all those edges (of the host graph) are shown that are incident to some assigned vertex.
• Figure 3.1: Running times for Dominating $M_3$-Pattern on different datasets in ms.
• Figure 3.2: Running times for Dominating $C_5$-Pattern (left) and $P_5$-Pattern on the stride dataset in ms.
• Figure 3.3: Influence of increasing the pattern size for matchings $M_3, M_4, M_5$. The left column is on the same dataset HoG-M3-5-hard. The generated OV instances in the right column increase in size with matching size.