Forbidden Subgraph Problems with Predictions

TL;DR

This work studies the Online Delayed Connected $H$-Node-Deletion problem, where a graph evolves vertex-by-vertex and must be kept $H$-free via irrevocable deletions. It introduces Alg$_p$, a prediction-aware online algorithm with counters that trade off trusting advice against robust fallback, and proves Pareto-optimality for broad classes of connected forbidden subgraphs $H$ (not containing two true twins or two false twins, and certain paths) by establishing explicit consistency-robustness bounds. The paper also proves lower bounds without advice and extended lower bounds with predictions, showing that for many $H$ the na"ive $k$-competitive baseline is optimal in the prediction-free setting and quantifies the impossibility of simultaneous improvements in consistency and robustness under untrusted predictions. It identifies subgraphs where better algorithms might exist (e.g., $P_3$-related structures) and outlines future work on refined prediction models and non-induced variants. Overall, the results illuminate when predictions help in online graph-deletion tasks and provide concrete Pareto-front algorithms and bounds that guide practical deployment.

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.

Figures (4)

• Figure 1: A gadget $g^i$ according to \ref{['lem:unadv_cl']} for $H = K_3$ and $i \neq 1$. On the left after presenting the first copy of $H$ and on the right after an algorithm deleted $v_1$ and $v_1'$ was reinserted. The vertices $v_1$ and $v_1'$ are in the same equivalence class $V^i_1$. Grey vertices indicate that they were deleted by the algorithm and the incident edges of deleted vertices are displayed as dashed.
• Figure 2: A gadget $g^i$ according to \ref{['lem:unadv_tr']} for $H = C_4$ and $i \neq 1$. On the left after the presentation of the first copy of $H$ and on the right after an algorithm deleted $v_1$ and $v_1'$ was reinserted. The vertices $v_1$ and $v_1'$ are in the same equivalence class $V^i_1$.
• Figure 3: A subgraph for which an algorithm with a better competitive ratio than $k=5$ might exist, that solves the Delayed Connected $H$-Node-Deletion Problem without advice.
• Figure 4: The combination of two gadgets $g^{i_1}$ and $g^{i_2}$ during a possible execution of an algorithm according to \ref{['thm:boundLowerTight']} for $H=P_3$. The advice of each vertex is given after its name and the corresponding gadget is given in the superscript of the name. Vertex ${v'}^{1,2}_1$ is part of both gadgets. Grey vertices indicate that they were deleted by the algorithm and the incident edges of deleted vertices are displayed as dashed. There are clearly induced paths of length three between both gadgets, e.g. $v^2_2, {v'}^{1,2}_1, v^1_1$.

Theorems & Definitions (20)

• Definition 1: Algorithm $A\hbox{LG}_p$
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Lemma 3
• proof
• Lemma 4