自适应随机图处理中的尖锐阈值

Pub Date : 2023-11-07 DOI:10.1002/rsa.21197
Calum MacRury, Erlang Surya
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引用次数: 0

摘要

这个过程是一个单人游戏,在这个游戏中,玩家最初看到的是一个空的顶点图。在每一步中,根据一个分布对边缘子集进行独立采样。然后玩家从中选择一条边,并将其添加到当前图形中。对于固定的单调递增图形属性,玩家的目标是迫使图形在尽可能少的步骤中得到满足。该过程推广了Achlioptas过程和半随机图过程。在此过程中,我们证明了一个尖锐阈值存在的充分条件。利用这个条件,我们证明了在半随机过程中,当对应于哈密顿量或包含完美匹配时,存在一个尖锐阈值。这解决了Ben - Eliezer等人(RSA, 2020)提出的两个开放性问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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Sharp thresholds in adaptive random graph processes
Abstract The ‐process is a single player game in which the player is initially presented the empty graph on vertices. In each step, a subset of edges is independently sampled according to a distribution . The player then selects one edge from , and adds to its current graph. For a fixed monotone increasing graph property , the objective of the player is to force the graph to satisfy in as few steps as possible. The ‐process generalizes both the Achlioptas process and the semi‐random graph process. We prove a sufficient condition for the existence of a sharp threshold for in the ‐process. Using this condition, in the semi‐random process we prove the existence of a sharp threshold when corresponds to being Hamiltonian or to containing a perfect matching. This resolves two of the open questions proposed by Ben‐Eliezer et al. (RSA, 2020).
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