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A linear time algorithm for linearizing quadratic and higher-order shortest path problems.

Eranda Çela1, Bettina Klinz1, Stefan Lendl2

  • 1Institute of Discrete Mathematics, Graz University of Technology, Graz, Austria.

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Summary
This summary is machine-generated.

Researchers developed a faster linear time algorithm for the Quadratic Shortest Path Problem (QSPP) on acyclic digraphs. This new method efficiently determines if a QSPP instance is linearizable, simplifying it to a classic Shortest Path Problem (SPP).

Keywords:
Higher-order shortest path problemLinearizationQuadratic shortest path problem

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Area of Science:

  • Discrete Mathematics
  • Theoretical Computer Science
  • Graph Theory

Background:

  • The Quadratic Shortest Path Problem (QSPP) is an NP-hard problem.
  • Linearizability of a QSPP instance means its equivalence to a classic Shortest Path Problem (SPP).
  • The linearization problem for QSPP (LinQSPP) identifies linearizable instances and their corresponding SPP.

Purpose of the Study:

  • To develop a novel, efficient algorithm for the LinQSPP on acyclic digraphs.
  • To improve upon existing algorithms for solving the LinQSPP.
  • To extend the findings to higher-order shortest path problems.

Main Methods:

  • A new linear time algorithm for LinQSPP on acyclic digraphs.
  • Leveraging a novel insight that linearizability is a local property for acyclic digraphs.
  • Algorithm design based on graph traversal and local property analysis.

Main Results:

  • A linear time algorithm for LinQSPP on acyclic digraphs, outperforming previous methods.
  • Demonstration that QSPP linearizability on acyclic digraphs is a local property.
  • The approach is extendable to higher-order shortest path problems.

Conclusions:

  • A significantly faster algorithm for LinQSPP on acyclic digraphs has been developed.
  • The local property insight simplifies the analysis and solution of QSPP linearization.
  • The presented method offers a more efficient approach to a class of shortest path problems.