The Theory of the Interleaving Distance on Multidimensional Persistence Modules

Michael Lesnick

doi:10.1007/s10208-015-9255-y

The Theory of the Interleaving Distance on Multidimensional Persistence Modules

Foundations of Computational Mathematics ◽

10.1007/s10208-015-9255-y ◽

2015 ◽

Vol 15 (3) ◽

pp. 613-650 ◽

Cited By ~ 31

Author(s):

Michael Lesnick

Keyword(s):

Multidimensional Persistence ◽

Persistence Modules ◽

Interleaving Distance

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Computing the Interleaving Distance is NP-Hard

Foundations of Computational Mathematics ◽

10.1007/s10208-019-09442-y ◽

2019 ◽

Vol 20 (5) ◽

pp. 1237-1271 ◽

Cited By ~ 3

Author(s):

Håvard Bakke Bjerkevik ◽

Magnus Bakke Botnan ◽

Michael Kerber

Keyword(s):

Computational Complexity ◽

Complete Characterization ◽

Np Hard ◽

Slight Improvement ◽

Approximation Factor ◽

Distance Computation ◽

Persistence Modules ◽

Np Complete ◽

Interleaving Distance

Abstract 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.

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