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The Steiner cycle and path cover problem on interval graphs.

Ante Ćustić1, Stefan Lendl2,3

  • 1Department of Mathematics, Simon Fraser University Surrey, 250-13450 102nd AV, Surrey, British Columbia V3T 0A3 Canada.

Journal of Combinatorial Optimization
|February 7, 2022
PubMed
Summary
This summary is machine-generated.

This study presents linear time algorithms for Steiner path cover and Steiner cycle problems on interval graphs. It proves that backward steps to non-Steiner intervals are unnecessary, optimizing pathfinding on these graph types.

Keywords:
Hamiltonian cycleInterval graphsLinear timeSteiner cycle

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Area of Science:

  • Graph Theory
  • Computational Complexity
  • Algorithm Design

Background:

  • The Steiner path problem generalizes Steiner tree and Hamiltonian path problems, focusing on paths visiting specific terminals.
  • The Steiner cycle problem seeks cycles through all terminals, while the Steiner path cover problem minimizes paths to cover all terminals.
  • Interval graphs are a special class of graphs with applications in various fields.

Purpose of the Study:

  • To develop efficient algorithms for the Steiner path cover and Steiner cycle problems on interval graphs.
  • To identify key properties of interval graphs that simplify these complex pathfinding problems.
  • To achieve linear time complexity for solving these problems on interval graphs.

Main Methods:

  • Developing a novel lemma proving backward steps to non-Steiner intervals are not required.
  • Adapting the deferred-query technique, originally by Chang et al., for interval graph problems.
  • Utilizing endpoint sorted lists for efficient representation and processing of interval graphs.

Main Results:

  • Linear time algorithms are presented for both the Steiner path cover and Steiner cycle problems on interval graphs.
  • The core lemma significantly simplifies the search space by eliminating unnecessary backward traversals.
  • The integration with the deferred-query technique ensures optimal computational efficiency.

Conclusions:

  • The study provides efficient, linear-time solutions for Steiner path cover and Steiner cycle problems on interval graphs.
  • The findings contribute to a deeper understanding of pathfinding algorithms on specialized graph classes.
  • The presented algorithms offer practical improvements for applications involving interval graphs.