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This study introduces a novel branch and cut approach for the traveling salesman problem (TSP). It leverages integer solutions and subtour elimination constraints for efficient optimization.

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Area of Science:

  • Combinatorial Optimization
  • Operations Research
  • Computer Science

Background:

  • The traveling salesman problem (TSP) is a fundamental challenge in combinatorial optimization.
  • Existing methods for solving TSP to optimality often involve relaxing integrality constraints and working with fractional solutions.
  • Branch and cut algorithms are standard for TSP, typically requiring complex separation procedures on fractional solutions.

Purpose of the Study:

  • To develop a novel branch and cut algorithm for the TSP that exclusively uses integer solutions.
  • To simplify the TSP solving process by avoiding fractional solutions and complex separation routines.
  • To enhance the efficiency of TSP solvers by focusing on subtour elimination constraints.

Main Methods:

  • A simplified integer linear programming (ILP) model is employed, relaxing only subtour elimination constraints.
  • The ILP model is solved to integer optimality, and violated subtour constraints are identified and added iteratively.
  • Techniques based on vertex clustering, empirical observations, and random graph theory are used to find relevant subtours efficiently.

Main Results:

  • The proposed method successfully solves the TSP using only integer solutions, bypassing fractional solution analysis.
  • Computational results on TSPLIB95 instances and random Euclidean graphs demonstrate the algorithm's performance.
  • The approach effectively identifies and incorporates subtour elimination constraints.

Conclusions:

  • The developed method offers a viable alternative to traditional branch and cut approaches for the TSP by utilizing integer solutions.
  • The strategy of relaxing only subtour elimination constraints proves effective for solving TSP instances.
  • Further research can explore advanced subtour identification strategies for enhanced performance.