Computational Complexity of the Interleaving Distance

The interleaving distance is arguably the most prominent distance measure in topological data analysis. In this paper, we provide bounds on the computational complexity of determining the interleaving distance in several settings. We show that the interleaving distance is NP-hard to compute for persistence modules valued in the category of vector spaces. In the specific setting of multidimensional persistent homology we show that the problem is at least as hard as a matrix invertibility problem. Furthermore, this allows us to conclude that the interleaving distance of interval decomposable modules depends on the characteristic of the field. Persistence modules valued in the category of sets are also studied. As a corollary, we obtain that the isomorphism problem for Reeb graphs is graph isomorphism complete.

[1]  Michael Lesnick,et al.  Multidimensional Interleavings and Applications to Topological Inference , 2012, ArXiv.

[2]  Eugene M. Luks,et al.  Testing isomorphism of modules , 2008 .

[3]  Afra Zomorodian,et al.  The Theory of Multidimensional Persistence , 2007, SCG '07.

[4]  Vin de Silva,et al.  Metrics for Generalized Persistence Modules , 2013, Found. Comput. Math..

[5]  Vin de Silva,et al.  Higher Interpolation and Extension for Persistence Modules , 2016, SIAM J. Appl. Algebra Geom..

[6]  Cary Webb.  Decomposition of graded modules , 1985 .

[7]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[8]  Håvard Bakke Bjerkevik.  Stability of higher-dimensional interval decomposable persistence modules , 2016, ArXiv.

[9]  Amit Patel,et al.  Categorified Reeb Graphs , 2015, Discret. Comput. Geom..

[10]  Dmitriy Morozov,et al.  Geometry Helps to Compare Persistence Diagrams , 2016, ALENEX.

[11]  Michael Lesnick,et al.  Algebraic Stability of Zigzag Persistence Modules , 2016, Algebraic & Geometric Topology.

[12]  Gunnar E. Carlsson,et al.  Zigzag Persistence , 2008, Found. Comput. Math..

[13]  Magnus Bakke Botnan.  Interval Decomposition of Infinite Zigzag Persistence Modules , 2015 .

[14]  W. Crawley-Boevey.  Decomposition of pointwise finite-dimensional persistence modules , 2012, 1210.0819.

[15]  Michael Lesnick,et al.  The Theory of the Interleaving Distance on Multidimensional Persistence Modules , 2011, Found. Comput. Math..

[16]  Wojciech Chachólski,et al.  Multidimensional Persistence and Noise , 2015, Foundations of Computational Mathematics.

[17]  Tamal K. Dey,et al.  Computing Bottleneck Distance for 2-D Interval Decomposable Modules , 2018, SoCG.

[18]  Kyle Fox,et al.  Computing the Gromov-Hausdorff Distance for Metric Trees , 2015, ISAAC.

[19]  Leonidas J. Guibas,et al.  Proximity of persistence modules and their diagrams , 2009, SCG '09.

[20]  Ulrich Bauer,et al.  Induced matchings and the algebraic stability of persistence barcodes , 2013, J. Comput. Geom..

[21]  Steve Oudot,et al.  Decomposition of Exact pfd Persistence Bimodules , 2020, Discret. Comput. Geom..

[22]  Primoz Skraba,et al.  Zigzag persistent homology in matrix multiplication time , 2011, SoCG '11.

[23]  Michael Barot,et al.  Introduction to the Representation Theory of Algebras , 2014 .

[24]  Regina Tyshkevich,et al.  Graph isomorphism problem , 1985 .

[25]  Daniela Giorgi,et al.  Multidimensional Size Functions for Shape Comparison , 2008, Journal of Mathematical Imaging and Vision.

[26]  Peter Bubenik,et al.  Categorification of Persistent Homology , 2012, Discret. Comput. Geom..

[27]  Oscar H. Ibarra,et al.  A Generalization of the Fast LUP Matrix Decomposition Algorithm and Applications , 1982, J. Algorithms.

[28]  Gorô Azumaya,et al.  Corrections and Supplementaries to My Paper concerning Krull-Remak-Schmidt’s Theorem , 1950, Nagoya Mathematical Journal.