SIAM Journal on Discrete Mathematics, Volume 34, Issue 3, Page 1854-1883, January 2020.
Consensus problems for strings and sequences appear in numerous application contexts, ranging from bioinformatics to data mining to machine learning. Closing some gaps in the literature, we show that several fundamental problems in this context are NP- and W[1]-hard and that the known (including some brute-force) algorithms are close to optimality assuming the Exponential Time Hypothesis. Among our main contributions is to settle the complexity status of computing a mean in dynamic time warping spaces which, as pointed out by Brill et al. [Data Min. Knowl. Discov., 33 (2019), pp. 252--291], suffered from many unproven or false assumptions in the literature. We prove this problem to be NP-hard and additionally show that a recent dynamic programming algorithm is essentially optimal. In this context, we study a broad family of circular string alignment problems. This family also serves as a key for our hardness reductions, and it is of independent (practical) interest in molecular biology. In particular, we show tight hardness and running time lower bounds for Circular Consensus String; notably, the corresponding noncircular version is easily linear-time solvable.

中文翻译：

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圆弦和时间序列共识问题的紧硬度结果

SIAM离散数学杂志，第34卷，第3期，第1854-1883页，2020年1月。
字符串和序列的共识性问题出现在许多应用程序上下文中，从生物信息学到数据挖掘再到机器学习。弥补文献中的一些空白，我们证明了在这种情况下的一些基本问题是NP-和W [1] -hard，并且已知的（包括一些蛮力）算法在假设指数时间假设的情况下接近最优。正如Brill等人所指出的，我们的主要贡献是解决了动态时间规整空间中计算平均值的复杂性状态。[数据最小值。知道 Discov。，33（2019），pp.252--291]，文献中有许多未经证实或错误的假设。我们证明这个问题是NP难的，另外还表明，最近的动态规划算法本质上是最优的。在这种情况下，我们研究了广泛的圆串对齐问题。这个家族也是我们降低硬度的关键，并且在分子生物学中具有独立的（实际的）兴趣。特别是，我们显示了圆形共识字符串的严格硬度和运行时间下限；值得注意的是，相应的非圆形版本很容易线性时间求解。