Motivated by alignment of correlated sparse random graphs, we introduce a hypothesis testing problem of deciding whether or not two random trees are correlated. We obtain sufficient conditions under which this testing is impossible or feasible. We propose MPAlign, a message-passing algorithm for graph alignment inspired by the tree correlation detection problem. We prove MPAlign 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 then conjecture that graph alignment is not feasible in polynomial time when the associated tree detection problem is impossible. If true, this conjecture together with our sufficient conditions on tree detection impossibility would imply the existence of a hard phase for graph alignment, i.e. a parameter range where alignment cannot be done in polynomial time even though it is known to be feasible in non-polynomial time.

翻译：以相关稀随机图的对齐为动力, 我们引入了一个假设测试问题, 以决定两个随机树是否相关。 我们获得了无法或可行的充分条件 。 我们提议了 MPAlign, 这是一种由树相关检测问题引发的图形对齐信息传递算法 。 我们证明 MPAlign 在树探测可行时, 可以在部分对齐时成功完成多数值对齐。 通过对树探测的分析, 我们发现新的参数范围, 在多元时间可以部分对齐稀随机图。 然后我们推测, 在相关树检测问题不可能发生时, 图形对齐在多数值时间是行不通的 。 如果确实如此, 这种预测加上树探测无法做到的充足条件, 将意味着图形对齐存在一个硬的阶段, 也就是说, 在一个多数值时间无法对齐的参数范围, 即使已知在非多数值时间是可行的 。