Forbidden Subgraph Problems with Predictions

TL;DR

This paper studies online algorithms for deleting vertices in graphs to avoid certain forbidden subgraphs, providing tight bounds and algorithms that incorporate predictions with robustness and optimality considerations.

Contribution

It introduces algorithms for forbidden subgraph problems with predictions, achieving Pareto-optimality in competitiveness and robustness for specific graph classes.

Findings

Tight bounds on competitive ratios for certain forbidden subgraphs.

Algorithms with predictions are Pareto-optimal for online vertex cover in specified cases.

Identification of subgraphs where better algorithms could be developed.

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.