arXiv:2611.06773 · v1

Approximation and Hardness for the Data Validation Problem

首次提交 2026-11-05 00:50

版本更新 2026-11-05 00:50

雷达观测 2026-11-05 12:13

摘要

Finnah, Gönsch and Otto [Eur. J. Oper. Res. 2025] recently introduced the Data Validation Problem (DVP): Given a transitive directed acyclic graph $G$ on $n$ vertices (resp. tasks) and $m$ edges (resp. precedences) and a probing budget, a policy can successively pay a probing fee and probe an edge in $G$ as long as the total probing fees do not exceed the budget. If a probe succeeds, the targeted edge $e$ is removed from $G$ and we receive a reward; if the probe fails, the edge stays in $G$ and cannot be probed again. We seek a policy with maximum expected cumulative reward. Crucially, $G$ must stay transitive throughout the probing process, so a policy can only challenge the precedence between consecutive tasks, that is, only probe edges in the transitive reduction of the current precedence graph. We show that DVP is strongly NP-hard and W[1]-hard with the probing budget $T$ as a parameter, even in the deterministic case (when every probe has probability $0$ or $1$ to succeed) and with unit probing costs and rewards. For the same setting, we give an approximation-preserving reduction from the Densest $k$-Subgraph (D$k$S) Problem, with the consequence that there is no polynomial-time $2^{(\log n)^{1-δ}}$-approximation algorithm and no FPT $\sqrt{T}$-approximation algorithm for DVP, assuming the Almost-Exponential Time Hypothesis and the Strongish Planted Clique Hypothesis, respectively. We also present the first known policies with provable worst-case guarantees for DVP, namely an $m$-approximation algorithm for instances with uniform probabilities and probing costs, a $2T/p_{\min}^T$-approximation algorithm for general instances, where $p_{\min}$ is the minimum probing success probability, and an exact FPT algorithm on tournament graphs, in the deterministic setting.