关于核化与计算无根一致森林的思考
Reflections on kernelizing and computing unrooted agreement forests
AI总结：
本文研究无根二元系统发育树的 TBR 距离/最大一致森林问题，实现并评估传统与更强核化规则，开发结合 ILP 的求解器 Tubro，证明新规则具有显著实际归约价值。
摘要：
系统发育树是用于模拟物种进化的叶标记树。本文探讨核化（即数据归约）对计算两棵无根二元系统发育树之间 TBR 距离这一 NP 难问题的实际影响。该问题在文献中更常被称为最大一致森林问题，其目标是将两棵树划分为数量最少的公共、互不重叠子树。我们实现了两条著名的归约规则，即子树归约和链归约，以及五条较新的、理论上更强的归约规则，并比较了使用与不使用这些更强规则时所获得的归约效果。我们发现，新规则会产生更小的归约实例，因此具有明显的实际附加价值。在许多情况下，它们还会以受控方式使 TBR 距离下降。接下来，我们将所实现的归约与已知最坏情形理论界进行比较：两棵归约树的叶数分别以 15k-9 和 11k-9 为界，其中 k 是 TBR 距离；我们在两种情况下都观察到，实际归约幅度远大于理论界。作为实验框架的副产品，我们获得了若干关于 TBR 距离实际计算的新认识。例如，我们发现通过随机采样某些精心构造的叶标签划分，可以高效获得非常强的 TBR 距离下界，并识别出一些似乎特别难以精确求解的实例。这些归约规则已在我们的新求解器 Tubro 中实现，该求解器将核化与整数线性规划（ILP）方法相结合。Tubro 还纳入了若干附加功能，例如簇归约和实用的上界启发式方法，并且能够利用归约规则正确性证明中产生的组合洞见来简化 ILP。
英文摘要：
Phylogenetic trees are leaf-labelled trees used to model the evolution of species. Here we explore the practical impact of kernelization (i.e. data reduction) on the NP-hard problem of computing the TBR distance between two unrooted binary phylogenetic trees. This problem is better-known in the literature as the maximum agreement forest problem, where the goal is to partition the two trees into a minimum number of common, non-overlapping subtrees. We have implemented two well-known reduction rules, the subtree and chain reduction, and five more recent, theoretically stronger reduction rules, and compare the reduction achieved with and without the stronger rules. We find that the new rules yield smaller reduced instances and thus have clear practical added value. In many cases they also cause the TBR distance to decrease in a controlled fashion. Next, we compare the achieved reduction to the known worst-case theoretical bounds of 15k-9 and 11k-9 respectively, on the number of leaves of the two reduced trees, where k is the TBR distance, observing in both cases a far larger reduction in practice. As a by-product of our experimental framework we obtain a number of new insights into the actual computation of TBR distance. We find, for example, that very strong lower bounds on TBR distance can be obtained efficiently by randomly sampling certain carefully constructed partitions of the leaf labels, and identify instances which seem particularly challenging to solve exactly. The reduction rules have been implemented within our new solver Tubro which combines kernelization with an Integer Linear Programming (ILP) approach. Tubro also incorporates a number of additional features, such as a cluster reduction and a practical upper-bounding heuristic, and it can leverage combinatorial insights emerging from the proofs of correctness of the reduction rules to simplify the ILP.
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