第三次打击完美的生育
A third strike against perfect phylogeny
摘要：
完美的进化树是研究进化树的基础，因为它们捕捉到了每一个进化特征在历史上只出现一次的情况;如果这样的事件被认为是罕见的，那么根据奥卡姆剃刀，这样的简约树更适合作为进化的假设。一个经典的结果指出，2状态字符允许一个完美的同源性，如果每个子集的2个字符允许一个。最近，它表明，对于3状态字符的相同属性，但大小为3的子集。一个长期存在的问题是，是否每个状态数都存在这样一个常数。更确切地说，已经证明，对于任何固定整数$r$，存在一个常数$f(r)$，使得一组$r$状态字符$C$具有完美的同源性，当且仅当至多$f(r)$字符的每个子集具有完美的同源性.在本文中，我们证明了这个猜想是错误的。特别是，我们表明，对于任何常数$t$，存在一组$C$的$8$状态字符，使$C$没有完美的同源性，但存在一个完美的同源性的$t$字符的每个子集。这个负面结果补充了Bodlaender等人的两个负面结果（“罢工”）。我们反思了第三次罢工的后果，指出虽然它确实关闭了一些有效算法开发的路线，但许多其他路线仍然是开放的。
英文摘要：
Perfect phylogenies are fundamental in the study of evolutionary trees because they capture the situation when each evolutionary trait emerges only once in history; if such events are believed to be rare, then by Occam's Razor such parsimonious trees are preferable as a hypothesis of evolution. A classical result states that 2-state characters permit a perfect phylogeny precisely if each subset of 2 characters permits one. More recently, it was shown that for 3-state characters the same property holds but for size-3 subsets. A long-standing open problem asked whether such a constant exists for each number of states. More precisely, it has been conjectured that for any fixed integer $r$, there exists a constant $f(r)$ such that a set of $r$-state characters $C$ has a perfect phylogeny if and only if every subset of at most $f(r)$ characters has a perfect phylogeny. In this paper, we show that this conjecture is false. In particular, we show that for any constant $t$, there exists a set $C$ of $8$-state characters such that $C$ has no perfect phylogeny, but there exists a perfect phylogeny for every subset of $t$ characters. This negative result complements the two negative results ("strikes") of Bodlaender et al. We reflect on the consequences of this third strike, pointing out that while it does close off some routes for efficient algorithm development, many others remain open.
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