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Random costs in combinatorial optimization

Mertens1

  • 1Institut fur Theoretische Physik, Universitat Magdeburg, Universitatsplatz 2, D-39106 Magdeburg, Germany.

Physical Review Letters
|October 4, 2000
PubMed
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The random cost problem, equivalent to NP-hard number partitioning, requires exhaustive search. This equivalence explains heuristic failures and enables calculating optimum cost probabilities.

Area of Science:

  • Computer Science
  • Optimization Theory
  • Computational Complexity

Background:

  • The random cost problem involves finding the minimum value in a list of random numbers.
  • This problem is inherently limited to exhaustive search for guaranteed optimal solutions.
  • Classical NP-hard problems often exhibit complex computational behaviors.

Purpose of the Study:

  • To establish the relationship between the random cost problem and the number partitioning problem.
  • To explain the limitations of heuristic algorithms in solving number partitioning.
  • To enable the calculation of probability distributions for optimal and near-optimal solutions.

Main Methods:

  • Demonstrating the essential equivalence between the random cost problem and number partitioning.

Related Experiment Videos

  • Analyzing the computational complexity implications of this equivalence.
  • Developing methods to compute probability distributions for costs.

    • Main Results:

    • The random cost problem is shown to be equivalent to the NP-hard number partitioning problem.
    • This equivalence explains the poor performance of heuristic methods on number partitioning.
    • The established equivalence facilitates the calculation of probability distributions for optimum and suboptimum costs.

    Conclusions:

    • The random cost problem's computational difficulty is directly linked to NP-hard problems like number partitioning.
    • Understanding this link provides insights into algorithm performance and limitations.
    • The findings enable probabilistic analysis of solutions for these optimization problems.