Motivated by alignment of correlated sparse random graphs, we study a hypothesis problem of deciding whether two random trees are correlated or not. Based on this tree detection problem, we propose BPAlign, a message-passing -- belief propagation -- algorithm for graph alignment, which we prove to succeed in polynomial time at partial alignment whenever tree detection is feasible. As a result our analysis of tree detection reveals new ranges of parameters for which partial alignment of sparse random graphs is feasible in polynomial time. We conjecture that the connection between partial graph alignment and tree detection runs deeper, and that the parameter range where tree detection is impossible, which we partially characterize, corresponds to a region where partial graph alignment is hard (not polytime feasible).

翻译：我们研究一个假设问题,即决定两棵随机树是否相关。基于此树探测问题,我们提议BPalign,一个信息传递 -- -- 信仰传播 -- -- 图表对齐算法,我们证明,只要树探测可行,在多角度对齐部分时间就能成功。结果,我们对树探测的分析揭示了新的参数范围,在多层次时间,稀散随机图的部分对齐是可行的。我们推测部分图形对齐和树探测之间的联系会更深,而部分图形对齐(不是多时间可行的)的参数范围与部分图形对齐(部分时间不可行)的地区相对应,我们部分特征是无法检测树木的参数范围。