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We present an optimal matching solution for assignment and matching problems with convex costs. Analytical results reveal how average costs scale with the number of points for power-law costs, confirmed by simulations.

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Area of Science:

  • Optimization theory
  • Computational mathematics
  • Statistical physics

Background:

  • The assignment and matching problems are fundamental in combinatorial optimization.
  • Understanding their behavior with convex cost functions is crucial for various applications.
  • Previous studies often focused on specific cost functions or limited domains.

Purpose of the Study:

  • To derive an optimal matching solution for one-dimensional assignment and matching problems.
  • To analyze the behavior of these problems for a broad class of convex cost functions.
  • To investigate the impact of random point distributions and asymptotic regimes on optimal costs.

Main Methods:

  • Analytical derivation of optimal matching solutions.
  • Asymptotic analysis for a large number of points (N).
  • Investigation of power-law cost functions c(z)=z^p for p>1 and p<0.
  • Comparison of theoretical predictions with numerical simulations.

Main Results:

  • Obtained analytical expressions for the average optimal cost in the asymptotic regime.
  • Determined the scaling of optimal mean cost with N: N^{-p/2} for assignment and N^{-p} for matching when p>1.
  • Found that the average optimal cost becomes constant for p<0 in both cases.
  • Validated theoretical findings through numerical simulations.

Conclusions:

  • The study provides a comprehensive analytical framework for one-dimensional assignment and matching problems with convex costs.
  • The derived scaling laws offer insights into the efficiency of matching algorithms as the number of points increases.
  • The findings are robust across different topological settings (interval and circumference) and for power-law cost functions.