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A note on computational approaches for the antibandwidth problem.

Markus Sinnl1,2

  • 1Department of Statistics and Operations Research, Economics and Statistics, Faculty of Business, University of Vienna, Vienna, Austria.

Central European Journal of Operations Research
|November 15, 2021
PubMed
Summary
This summary is machine-generated.

This study addresses the antibandwidth problem, aiming to find optimal graph labelings. New algorithms successfully solved eight previously unsolved instances and improved solutions for eleven more, significantly reducing optimality gaps.

Keywords:
BandwidthClique problemConstraint programmingGraph labelingInteger programming

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Area of Science:

  • Graph Theory
  • Combinatorial Optimization
  • Computer Science

Background:

  • The antibandwidth problem, also known as the dual bandwidth or maximum differential coloring problem, seeks to maximize the minimum difference between labels of adjacent vertices in a graph.
  • This NP-hard problem has applications in scheduling, radio frequency assignment, and facility location.
  • Existing methods struggle with benchmark instances (HarwellBoeing), showing large optimality gaps and few solved cases.

Purpose of the Study:

  • To develop novel algorithms for solving the antibandwidth problem more effectively.
  • To improve upon existing upper bounds and find proven optimal solutions for challenging instances.
  • To reduce the significant optimality gaps observed in current literature.

Main Methods:

  • Development of new mixed-integer programming (MIP) models and valid inequalities.
  • Design of a branch-and-cut algorithm and an iterative solution algorithm.
  • Implementation of constraint programming approaches and calculation of upper bounds using stability and chromatic numbers.

Main Results:

  • Proven optimal solutions were found for eight previously unsolved instances.
  • Optimality gaps were reduced for eleven additional instances.
  • Improved solution values were achieved for seven instances, with the largest remaining gap reduced to 46%.

Conclusions:

  • The proposed MIP models and algorithms significantly advance the state-of-the-art in solving the antibandwidth problem.
  • The methods demonstrate effectiveness on challenging benchmark instances, overcoming limitations of previous approaches.
  • This work provides a more robust framework for tackling complex graph labeling problems.