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The complexity of divisibility.

Johannes Bausch1, Toby Cubitt1,2

  • 1DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK.

Linear Algebra and Its Applications
|June 20, 2017
PubMed
Summary
This summary is machine-generated.

Finite divisibility of stochastic matrices is NP-complete, while probability distribution divisibility is in P but decomposability is NP-hard. These findings resolve long-standing questions in linear algebra and probability theory.

Keywords:
60-0868Q3081-08Complexity theoryDecomposabilityDivisibilityProbability distributionsStochastic matricescptp maps

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Area of Science:

  • Linear Algebra
  • Probability Theory
  • Computational Complexity

Background:

  • Addresses two long-standing open questions regarding stochastic matrices and probability distributions.
  • Focuses on computational complexity aspects of divisibility and decomposability.

Purpose of the Study:

  • To determine the computational complexity of finite stochastic matrix divisibility.
  • To analyze the complexity of divisibility and decomposability for probability distributions.
  • To extend findings to nonnegative matrices and quantum analogues of stochastic matrices.

Main Methods:

  • Utilizes computational complexity theory to analyze the problems.
  • Develops a polynomial-time algorithm for finite distribution divisibility.
  • Proves NP-completeness for stochastic matrix divisibility and NP-hardness for distribution decomposability.

Main Results:

  • Finite stochastic matrix divisibility is proven to be NP-complete.
  • This result extends to nonnegative matrices and completely-positive trace-preserving maps.
  • A complexity hierarchy for probability distributions is established: finite divisibility is in P, while decomposability is NP-hard.
  • Results hold for weak-membership formulations, indicating robustness to perturbations.

Conclusions:

  • Resolves fundamental questions in linear algebra and probability theory from a computational perspective.
  • Establishes clear complexity boundaries for matrix and distribution divisibility/decomposability.
  • Provides an efficient algorithm for a key distribution problem.