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Statistical mechanical analysis of linear programming relaxation for combinatorial optimization problems.

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  • 1Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan.

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Linear programming (LP) relaxation for minimum vertex cover (min-VC) problems accurately estimates integer programming (IP) solutions below a critical graph density. This statistical mechanics approach reveals when LP relaxation is effective for complex graph problems.

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Area of Science:

  • Statistical Mechanics
  • Computer Science
  • Discrete Mathematics

Background:

  • The minimum vertex cover (min-VC) problem is a fundamental combinatorial optimization problem.
  • Linear programming (LP) is often used as a relaxation technique for integer programming (IP) problems like min-VC.
  • Understanding the accuracy of LP relaxation in random graph settings is crucial for theoretical and practical applications.

Purpose of the Study:

  • To investigate the typical behavior of LP as a relaxation for the min-VC problem on random graphs.
  • To propose a unified statistical mechanical model for both LP and IP formulations of min-VC.
  • To determine the conditions under which LP relaxation provides accurate solutions for min-VC and related problems.

Main Methods:

  • A lattice-gas model is employed on Erdös-Rényi random graphs with α-uniform hyperedges.
  • Statistical mechanical analyses are performed to study the LP and IP problems within a single-parameter family.
  • Analytic and numerical methods are used to evaluate the accuracy of LP relaxation.

Main Results:

  • For α=2, the LP optimal solution matches the IP solution below a critical average degree (c=e) in the thermodynamic limit.
  • The accuracy threshold for LP relaxation (c=e) extends the known mathematical result (c=1) and aligns with the replica symmetry-breaking threshold for IP.
  • For α≥3, LP relaxation is found to be inaccurate above a critical average degree (c=e/(α-1)), coinciding with replica symmetry breaking.

Conclusions:

  • LP relaxation is a reliable approximation for min-VC on random graphs below specific critical densities.
  • The accuracy of LP relaxation is intrinsically linked to the replica symmetry-breaking properties of the corresponding IP problem.
  • The proposed statistical mechanical framework provides a unified approach to analyze LP/IP relaxations in graph problems.