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This study introduces new integer programming formulations and branch-and-cut algorithms for the discrete k-neighbor k-center problem (d-k-k-CP). These methods effectively solve many problem instances to proven optimality, improving existing solutions.

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Area of Science:

  • Operations Research
  • Discrete Optimization
  • Facility Location Problems

Background:

  • The discrete k-neighbor k-center problem (d-k-k-CP) is a recently studied variant of the classical k-center problem.
  • Existing solutions for d-k-k-CP primarily consist of approximation algorithms and heuristics.

Purpose of the Study:

  • To develop exact algorithms for the d-k-k-CP.
  • To introduce novel integer programming formulations and associated theoretical results.

Main Methods:

  • Developed two integer programming formulations for the d-k-k-CP.
  • Incorporated lifting of inequalities, valid inequalities, and variable fixing procedures.
  • Constructed enhanced branch-and-cut (B&C) algorithms with starting and primal heuristics.

Main Results:

  • Achieved proven optimality for 116 out of 194 instances from literature.
  • Demonstrated effectiveness with most solved instances under a minute runtime.
  • Provided improved solution values for 116 instances.

Conclusions:

  • The proposed integer programming formulations and B&C algorithms are effective for solving the d-k-k-CP.
  • This work advances the state-of-the-art in solving this emerging facility location problem.