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Physicist's view on the unbalanced k-cardinality assignment problem.

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This study introduces a novel statistical physics approach to solve the computationally intensive k-cardinality assignment problem. The method offers an efficient and scalable solution, outperforming existing algorithms for complex assignment scenarios.

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Area of Science:

  • Operations Research
  • Statistical Physics
  • Computer Science

Background:

  • The k-cardinality assignment problem is crucial in resource allocation but computationally expensive with exact algorithms for large instances.
  • Existing exact methods become prohibitive as the number of agents (N) and tasks (M) increase beyond k.

Purpose of the Study:

  • To develop an efficient and scalable method for solving the k-cardinality unbalanced assignment problem.
  • To adapt techniques from statistical physics for a novel approach to assignment problems.

Main Methods:

  • Formulation of the k-cardinality assignment problem using statistical physics principles.
  • Derivation of a free-energy function with temperature annealing for optimization.
  • Development of a GPU-accelerated implementation using CUDA.

Main Results:

  • A strongly concave free-energy function was derived, monotonically decreasing to the optimal assignment cost.
  • The exact solution can be obtained via simple rounding for large inverse temperatures (β).
  • The GPU implementation demonstrates efficiency comparable to state-of-the-art parallel Hungarian algorithms and significantly faster for pathological cases.

Conclusions:

  • The statistical physics framework provides a robust and efficient method for the k-cardinality assignment problem.
  • The approach is adaptable to degenerate cases and offers significant speedups on parallel architectures.
  • This method presents a viable alternative to traditional algorithms, especially for large-scale and complex assignment problems.