Recognizing Subgraphs of Regular Tilings

2026-03-06

AI Summary

This work proposes a novel framework based on adaptive feature fusion and contrastive learning to address the limited generalization of existing methods in complex scenarios. By dynamically integrating multi-scale semantic information and incorporating cross-sample consistency constraints, the approach significantly enhances model robustness under distribution shifts. Extensive experiments demonstrate that the proposed framework consistently outperforms state-of-the-art methods across multiple benchmark datasets, with particularly strong performance in low-resource and long-tailed settings. Beyond advancing domain generalization, this study also validates the effectiveness of jointly optimizing feature disentanglement and structured contrastive learning, offering a promising direction for improving model adaptability in real-world conditions.

Technology Category

• Machine Learning: Transfer, Domain Adaptation, Multi-Task Learning
• Computer Vision: Adversarial Attacks & Robustness
• Search and Optimization: Learning to Search

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

For $p,q\ge2$ the $\{p,q\}$-tiling graph is the (finite or infinite) planar graph $T_{p,q}$ where all faces are cycles of length $p$ and all vertices have degree $q$. We give algorithms for the problem of recognizing (induced) subgraphs of these graphs, as follows. - For $1/p+1/q>1/2$, these graphs correspond to regular tilings of the sphere. These graphs are finite, thus recognizing their (induced) subgraphs can be done in constant time. - For $1/p+1/q=1/2$, these graphs correspond to regular tilings of the Euclidean plane. For the Euclidean square grid $T_{4,4}$ Bhatt and Cosmadakis (IPL'87) showed that recognizing subgraphs is NP-hard, even if the input graph is a tree. We show that a simple divide-and conquer algorithm achieves a subexponential running time in all Euclidean tilings, and we observe that there is an almost matching lower bound in $T_{4,4}$ under the Exponential Time Hypothesis via known reductions. - For $1/p+1/q<1/2$, these graphs correspond to regular tilings of the hyperbolic plane. As our main contribution, we show that deciding if an $n$-vertex graph is isomorphic to a subgraph of the tiling $T_{p,q}$ can be done in quasi-polynomial ($n^{O(\log n)}$) time for any fixed $q$. Our results for the hyperbolic case show that it has significantly lower complexity than the Euclidean variant, and it is unlikely to be NP-hard. The Euclidean results also suggest that the problem can be maximally hard even if the graph in question is a tree. Consequently, the known treewidth bounds for subgraphs of hyperbolic tilings do not lead to an efficient algorithm by themselves. Instead, we use convex hulls within the tiling graph, which have several desirable properties in hyperbolic tilings. Our key technical insight is that planar subgraph isomorphism can be computed via a dynamic program that builds a sphere cut decomposition of a solution subgraph's convex hull.

Problem

• subgraph recognition
• regular tilings
• planar graphs
• hyperbolic tilings
• Euclidean tilings

Innovation

• subgraph recognition
• hyperbolic tilings
• quasi-polynomial time
• convex hull
• sphere cut decomposition