The Fixed-Tree BMEP (FT-BMEP) is a special case of the Balanced Minimum Evolution Problem (BMEP) that consists of finding the assignment of a set of n taxa to the n leaves of a given unrooted binary tree so as to minimize the BMEP objective function. Deciding the computational complexity of the FT-BMEP has been an open problem for almost a decade. Here, we show that a few modifications to Fiorini and Joret’s proof of the \(\mathcal {NP}\)-hardness of the BMEP suffice to prove the general \(\mathcal {NP}\)-hardness of the FT-BMEP as well as its strong inapproximability.

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关于固定树平衡最小进化问题的逼近性

固定树 BMEP (FT-BMEP)是平衡最小进化问题(BMEP)的一个特例，它包括找到一组n类群到给定无根二叉树的n 个叶子的分配，以最小化BMEP 目标函数。近十年来，确定 FT-BMEP 的计算复杂度一直是一个悬而未决的问题。在这里，我们展示了对 Fiorini 和 Joret 证明 BMEP 的\(\mathcal {NP}\)硬度的一些修改足以证明FT-BMEP的一般\(\mathcal {NP}\)硬度以及它强大的不可近似性。