断点距离下重排问题的复杂性
On the Complexity of Rearrangement Problems under the Breakpoint Distance
摘要：
研究了广义断点模型中重排问题的复杂性，解决了几个悬而未决的问题。该模型由Tannier等人（2009年）引入，他们表明在多染色体循环和混合断点模型中，中位数问题在多项式时间内可解。这是有趣的，因为在大多数其他重排模型（DCJ，逆转，单染色体或多线性断点模型），这个问题是NP难的。断点距离下的小甚至大的遗传问题的复杂性仍然是一个悬而未决的问题。我们改进了算法的中位数问题，并表明它是等价的问题，找到最大基数非二分匹配（线性约简下）。另一方面，我们证明了更一般的小规模遗传问题是NP-困难的。令人惊讶的是，我们表明，它已经是NP-硬（甚至APX-硬）的4个物种（一个四重共生）。换句话说，虽然找到3个物种的祖先很容易，但找到4个物种的两个祖先已经很难了。我们还表明，在unichromosomal和多线性断点模型，减半问题是NP-难的，从而驳斥了Tannier等人的猜想。有趣的是，这是第一个问题，这是更难的断点模型比在DCJ或逆转模型。
英文摘要：
We study complexity of rearrangement problems in the generalized breakpoint model and settle several open questions. The model was introduced by Tannier et al. (2009) who showed that the median problem is solvable in polynomial time in the multichromosomal circular and mixed breakpoint models. This is intriguing, since in most other rearrangement models (DCJ, reversal, unichromosomal or multilinear breakpoint models), the problem is NP-hard. The complexity of the small or even the large phylogeny problem under the breakpoint distance remained an open problem. We improve the algorithm for the median problem and show that it is equivalent to the problem of finding maximum cardinality non-bipartite matching (under linear reduction). On the other hand, we prove that the more general small phylogeny problem is NP-hard. Surprisingly, we show that it is already NP-hard (or even APX-hard) for 4 species (a quartet phylogeny). In other words, while finding an ancestor for 3 species is easy, finding two ancestors for 4 species is already hard. We also show that, in the unichromosomal and the multilinear breakpoint model, the halving problem is NP-hard, thus refuting the conjecture of Tannier et al. Interestingly, this is the first problem which is harder in the breakpoint model than in the DCJ or reversal models.
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