Fine-Grained Classification Of Detecting Dominating Patterns

2025-09-26

AI Summary

This paper studies the detection of “dominating P-patterns” in graphs: finding a vertex subset that is both a dominating set and induces a subgraph isomorphic to a given pattern graph P. For all pattern graphs P, we introduce a novel graph parameter ρ(P) and, assuming ω = 2, achieve the first fine-grained complexity classification for all P except triangles. Leveraging the Orthogonal Vectors Hypothesis (OVH), matrix multiplication exponent analysis, parameterized complexity, and induced subgraph counting techniques, we establish tight upper and lower bounds, yielding a conditionally optimal time bound of $(n^{ ho(P)} m^{(|V(P)|- ho(P))/2})^{1pm o(1)}$. Our work establishes the first systematic fine-grained complexity framework for dominating pattern detection, revealing the intrinsic computational hardness arising from the coupling of domination constraints and structural isomorphism requirements.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider the following generalization of dominating sets: Let $G$ be a host graph and $P$ be a pattern graph $P$. A dominating $P$-pattern in $G$ is a subset $S$ of vertices in $G$ that (1) forms a dominating set in $G$ emph{and} (2) induces a subgraph isomorphic to $P$. The graph theory literature studies the properties of dominating $P$-patterns for various patterns $P$, including cliques, matchings, independent sets, cycles and paths. Previous work (Kunnemann, Redzic 2024) obtains algorithms and conditional lower bounds for detecting dominating $P$-patterns particularly for $P$ being a $k$-clique, a $k$-independent set and a $k$-matching. Their results give conditionally tight lower bounds if $k$ is sufficiently large (where the bound depends the matrix multiplication exponent $ω$). We ask: Can we obtain a classification of the fine-grained complexity for emph{all} patterns $P$? Indeed, we define a graph parameter $ρ(P)$ such that if $ω=2$, then [ left(n^{ρ(P)} m^{frac{|V(P)|-ρ(P)}{2}} ight)^{1pm o(1)} ] is the optimal running time assuming the Orthogonal Vectors Hypothesis, for all patterns $P$ except the triangle $K_3$. Here, the host graph $G$ has $n$ vertices and $m=Θ(n^α)$ edges, where $1le αle 2$. The parameter $ρ(P)$ is closely related (but sometimes different) to a parameter $δ(P) = max_{Ssubseteq V(P)} |S|-|N(S)|$ studied in (Alon 1981) to tightly quantify the maximum number of occurrences of induced subgraphs isomorphic to $P$. Our results stand in contrast to the lack of a full fine-grained classification of detecting an arbitrary (not necessarily emph{dominating}) induced $P$-pattern.

Problem

• Classifying fine-grained complexity for dominating pattern detection in graphs
• Determining optimal running time for all patterns except triangles
• Defining graph parameter ρ(P) to characterize computational complexity

Innovation

• Defining graph parameter ρ(P) for classification
• Deriving optimal runtime using Orthogonal Vectors Hypothesis
• Extending beyond triangle K3 for all patterns