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Exact satisfiability threshold for k-satisfiability problems on a Bethe lattice.

Supriya Krishnamurthy1, Sumedha2

  • 1Department of Physics, Stockholm University, SE 106 91, Stockholm, Sweden.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 14, 2015
PubMed
Summary
This summary is machine-generated.

We present a new method to find the satisfiability threshold for constraint satisfaction problems on Bethe lattices without assuming solution-space structure. This approach matches existing predictions and offers new interpretations for replica-symmetry-breaking methods.

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Area of Science:

  • Statistical physics
  • Theoretical computer science
  • Constraint satisfaction problems

Background:

  • The satisfiability threshold is crucial for understanding the solvability of constraint satisfaction problems.
  • Existing methods like moment methods provide rigorous bounds, while replica-symmetry-breaking (RSB) offers accurate predictions but relies on assumptions about solution structure.

Purpose of the Study:

  • To develop a novel, assumption-free method for determining the satisfiability threshold on Bethe lattices.
  • To provide alternative interpretations and motivations for existing replica-symmetry-breaking (RSB) equations.

Main Methods:

  • Developed a new analytical route to calculate the satisfiability threshold on a Bethe lattice.
  • Avoided making assumptions about the underlying solution-space structure, a limitation of the RSB approach.

Main Results:

  • The derived expressions and threshold values precisely match the predictions of the cavity method under the one-step RSB hypothesis.
  • The method successfully calculates other quantities, such as the second moment of the number of solutions on a Bethe lattice.

Conclusions:

  • The novel method provides a rigorous and assumption-free way to determine satisfiability thresholds and related quantities.
  • This work offers new insights into the replica-symmetry-breaking (RSB) approach and its underlying equations.