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Clique Homology is -hard.

Marcos Crichigno1, Tamara Kohler2,3

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Determining homology groups of simplicial complexes is computationally hard, with decision problems being PSPACE-hard and counting problems being #P-hard. This complexity holds even for clique complexes relevant to topological data analysis.

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Area of Science:

  • Computational Topology
  • Computational Complexity Theory
  • Algebraic Topology

Background:

  • The computational complexity of determining homology groups of simplicial complexes has been a long-standing open question for over two decades.
  • Simplicial complexes are fundamental structures in computational topology and topological data analysis.

Purpose of the Study:

  • To resolve the computational complexity of determining homology groups of simplicial complexes.
  • To investigate the hardness of these problems for clique complexes, a relevant family for topological data analysis.

Main Methods:

  • The study combines techniques from Hamiltonian complexity and algebraic topology.
  • Complexity analysis was performed on decision and exact counting versions of the problem.

Main Results:

  • The decision problem for homology groups of simplicial complexes is PSPACE-hard.
  • The exact counting version of the problem is #P-hard.
  • These hardness results hold even for clique complexes.

Conclusions:

  • The computational complexity of homology groups suggests a potential quantum mechanical nature for these problems under certain constraints.
  • The findings have implications for understanding quantum advantage in topological data analysis.