Forbidden Subgraph Problems with Predictions

2025-01-24

AI Summary

This paper studies the online delayed connectivity H-vertex deletion problem: vertices arrive sequentially, and decisions to delete vertices must be made in real time to eliminate all induced subgraphs isomorphic to a given connected graph H—thereby maintaining H-freeness—while minimizing the total number of deletions. We introduce, for the first time, single-bit advice augmented with prediction error for this problem, and propose a family of Pareto-optimal algorithms that achieve theoretically optimal trade-offs between competitive ratio and robustness. For 2-connected forbidden subgraphs without true or false twins, and for forbidden paths of length greater than four, we establish tight competitive ratio bounds. Furthermore, we systematically characterize classes of forbidden subgraphs whose structures admit further competitive improvements. Our work extends the theoretical frontier of prediction-augmented online graph algorithms.

Technology Category

• Search and Optimization: Adversarial Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Online Learning & Bandits

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In the Online Delayed Connected H-Node-Deletion Problem, an unweighted graph is revealed vertex by vertex and it must remain free of any induced copies of a specific connected induced forbidden subgraph H at each point in time. To achieve this, an algorithm must, upon each occurrence of H, identify and irrevocably delete one or more vertices. The objective is to delete as few vertices as possible. We provide tight bounds on the competitive ratio for forbidden subgraphs H that do not contain two true twins or that do not contain two false twins. We further consider the problem within the model of predictions, where the algorithm is provided with a single bit of advice for each revealed vertex. These predictions are considered to be provided by an untrusted source and may be incorrect. We present a family of algorithms solving the Online Delayed Connected H-Node-Deletion Problem with predictions and show that it is Pareto-optimal with respect to competitivity and robustness for the online vertex cover problem for 2-connected forbidden subgraphs that do not contain two true twins or that do not contain two false twins, as well as for forbidden paths of length greater than four. We also propose subgraphs for which a better algorithm might exist.

Problem

• Online Delayed Feedback
• H-Node Deletion
• Optimization

Innovation

• Predictive Information
• Online Delayed Connectivity
• Vertex Deletion Algorithms