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Computing the Interleaving Distance is NP-Hard
Foundations of Computational Mathematics ( IF 3 ) Pub Date : 2019-11-11 , DOI: 10.1007/s10208-019-09442-y
Håvard Bakke Bjerkevik , Magnus Bakke Botnan , Michael Kerber

We show that computing the interleaving distance between two multi-graded persistence modules is NP-hard. More precisely, we show that deciding whether two modules are 1-interleaved is NP-complete, already for bigraded, interval decomposable modules. Our proof is based on previous work showing that a constrained matrix invertibility problem can be reduced to the interleaving distance computation of a special type of persistence modules. We show that this matrix invertibility problem is NP-complete. We also give a slight improvement in the above reduction, showing that also the approximation of the interleaving distance is NP-hard for any approximation factor smaller than 3. Additionally, we obtain corresponding hardness results for the case that the modules are indecomposable, and in the setting of one-sided stability. Furthermore, we show that checking for injections (resp. surjections) between persistence modules is NP-hard. In conjunction with earlier results from computational algebra this gives a complete characterization of the computational complexity of one-sided stability. Lastly, we show that it is in general NP-hard to approximate distances induced by noise systems within a factor of 2.



中文翻译:

计算交织距离是NP-Hard

我们表明,计算两个多级持久性模块之间的交织距离是NP难的。更准确地说,我们已经证明,对于大幅度的区间可分解模块,确定两个模块是否进行1次交错是NP完全的。我们的证明基于先前的工作,表明约束矩阵的可逆性问题可以简化为特殊类型的持久性模块的交织距离计算。我们证明这个矩阵可逆性问题是NP完全的。我们还对上述减小量做了一点改进,表明对于任何小于3的逼近因子,交织距离的逼近也是NP-hard。此外,在模块不可分解的情况下,我们可以获得相应的硬度结果。单边稳定性的设置。此外,我们表明,检查持久性模块之间的注入(重复排斥)是NP难的。结合计算代数的早期结果,这可以完全表征单边稳定性的计算复杂性。最后,我们表明,一般来说,NP很难将噪声系统引起的距离近似为2倍。

更新日期:2019-11-11
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