有向时间探索问题

中文摘要 AI 辅助

我们研究了时间有向图上的时间探索问题。我们证明了寿命为 $O(n^2)$ 足以保证在始终单边的时间有向图上存在时间探索。我们通过一个 $\Omega(n^2)$ 的下界来补充这一结果，即使在每个快照的最大无向度为 2 的情况下也是如此；对于始终强连通的时间有向图，即使最大无向度为 3，下界仍然成立。这与无向情形形成鲜明对比。对于大最小度情形，我们证明了寿命为 $4n/3 - 1$ 是充分且必要的，以保证每个快照都是半完全的时间有向图上存在时间探索。对于每个快照的最小无向度至少为 $n - c - 1$ 的始终强连通时间有向图，我们证明了寿命为 $O(cn)$ 保证存在时间探索，并且我们还证明了这是渐近紧的。从计算角度来看，我们关于时间半完全有向图的结果也产生了一个多项式时间、因子为 $4/3$ 的算法，用于判断一个时间半完全有向图是否在前 $\ell$ 个快照内允许时间探索。我们补充说明，除非 P$=$NP，否则不存在多项式时间、因子为 $(4/3 - \epsilon)$ 的近似算法，即使每个快照都是锦标赛图也是如此。

英文摘要

We study the temporal exploration problem on temporal digraphs. We prove that a lifetime of $O(n^2)$ suffices to guarantee the existence of a temporal exploration on always-unilateral temporal digraphs. We complement this with a $Ω(n^2)$ lower bound, even in the case where each snapshot has maximum undirected degree 2; for always-strong temporal digraphs, the lower bound still holds even if the maximum undirected degree is 3. This stands in stark contrast with the undirected setting. For the large minimum degree setting, we show that a lifetime of $4n/3 - 1$ is sufficient and necessary for guaranteeing the existence of a temporal exploration on temporal digraphs where each snapshot is semicomplete. For always-strong temporal digraphs where each snapshot has minimum undirected degree at least $n - c - 1$, we prove that a lifetime of $O(cn)$ guarantees the existence of a temporal exploration, and we also prove that this is asymptotically tight. From a computational perspective, our results for temporal semicomplete digraphs also yield a polynomial-time, factor-$4/3$ algorithm for deciding if a temporal semicomplete digraph admits a temporal exploration within the first $\ell$ snapshots. We complement this showing that no polynomial-time, factor-$(4/3 - ε)$ approximation algorithm exists, even if every snapshot is a tournament, unless P$=$NP.

发表机构

• National Institute of Informatics（日本国立情报学研究所）
• Fudan University（复旦大学）

机构由 AI 辅助整理，请以论文原文为准。