Reconfiguring Shortest Paths in Graphs

Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) repaving road networks, (b) rerouting data packets in a s...

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Published in	Algorithmica Vol. 86; no. 10; pp. 3309 - 3338
Main Authors	Gajjar, Kshitij, Jha, Agastya Vibhuti, Kumar, Manish, Lahiri, Abhiruk
Format	Journal Article
Language	English
Published	New York Springer US 01.10.2024 Springer Nature B.V
Subjects	Algorithm Analysis and Problem Complexity Algorithms Apexes Cargo containers Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Multiprocessing Packets (communication) Polynomials Roads & highways Shortest-path problems Stowage (onboard equipment) Theory of Computation Bridged graph Line graph Hardness of approximation Shortest path PSPACE-complete Circle graph Reconfiguration Boolean hypercube
Online Access	Get full text
ISSN	0178-4617 1432-0541 1432-0541
DOI	10.1007/s00453-024-01263-y

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Abstract	Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) repaving road networks, (b) rerouting data packets in a synchronous multiprocessing setting, (c) the shipping container stowage problem, and (d) the train marshalling problem. When modelled as graph problems, (a) is the most general case while (b), (c), (d) are restrictions to different graph classes. We show that (a) does not admit polynomial-time algorithms (assuming P ≠ NP ), even for relaxed variants of the problem (assuming P ≠ PSPACE ). For (b), (c), (d), we present polynomial-time algorithms to solve the respective problems. We also generalize the problem to when at most k (for a fixed integer k ≥ 2 ) contiguous vertices on a shortest path can be changed at a time.
AbstractList	Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) repaving road networks, (b) rerouting data packets in a synchronous multiprocessing setting, (c) the shipping container stowage problem, and (d) the train marshalling problem. When modelled as graph problems, (a) is the most general case while (b), (c), (d) are restrictions to different graph classes. We show that (a) does not admit polynomial-time algorithms (assuming P ≠ NP ), even for relaxed variants of the problem (assuming P ≠ PSPACE ). For (b), (c), (d), we present polynomial-time algorithms to solve the respective problems. We also generalize the problem to when at most k (for a fixed integer k ≥ 2 ) contiguous vertices on a shortest path can be changed at a time.Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) repaving road networks, (b) rerouting data packets in a synchronous multiprocessing setting, (c) the shipping container stowage problem, and (d) the train marshalling problem. When modelled as graph problems, (a) is the most general case while (b), (c), (d) are restrictions to different graph classes. We show that (a) does not admit polynomial-time algorithms (assuming P ≠ NP ), even for relaxed variants of the problem (assuming P ≠ PSPACE ). For (b), (c), (d), we present polynomial-time algorithms to solve the respective problems. We also generalize the problem to when at most k (for a fixed integer k ≥ 2 ) contiguous vertices on a shortest path can be changed at a time. Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) repaving road networks, (b) rerouting data packets in a synchronous multiprocessing setting, (c) the shipping container stowage problem, and (d) the train marshalling problem. When modelled as graph problems, (a) is the most general case while (b), (c), (d) are restrictions to different graph classes. We show that (a) does not admit polynomial-time algorithms (assuming P ≠ NP ), even for relaxed variants of the problem (assuming P ≠ PSPACE ). For (b), (c), (d), we present polynomial-time algorithms to solve the respective problems. We also generalize the problem to when at most k (for a fixed integer k ≥ 2 ) contiguous vertices on a shortest path can be changed at a time. Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) repaving road networks, (b) rerouting data packets in a synchronous multiprocessing setting, (c) the shipping container stowage problem, and (d) the train marshalling problem. When modelled as graph problems, (a) is the most general case while (b), (c), (d) are restrictions to different graph classes. We show that (a) does not admit polynomial-time algorithms (assuming $${{\,\mathrm{\texttt {P}}\,}}\ne {{\,\mathrm{\texttt {NP}}\,}}$$ P≠NP), even for relaxed variants of the problem (assuming $${{\,\mathrm{\texttt {P}}\,}}\ne {{\,\mathrm{\texttt {PSPACE}}\,}}$$ P≠PSPACE). For (b), (c), (d), we present polynomial-time algorithms to solve the respective problems. We also generalize the problem to when at most k (for a fixed integer $$k\ge 2$$ k≥2) contiguous vertices on a shortest path can be changed at a time. Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) repaving road networks, (b) rerouting data packets in a synchronous multiprocessing setting, (c) the shipping container stowage problem, and (d) the train marshalling problem. When modelled as graph problems, (a) is the most general case while (b), (c), (d) are restrictions to different graph classes. We show that (a) does not admit polynomial-time algorithms (assuming P≠NP), even for relaxed variants of the problem (assuming P≠PSPACE). For (b), (c), (d), we present polynomial-time algorithms to solve the respective problems. We also generalize the problem to when at most k (for a fixed integer k≥2) contiguous vertices on a shortest path can be changed at a time. Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) repaving road networks, (b) rerouting data packets in a synchronous multiprocessing setting, (c) the shipping container stowage problem, and (d) the train marshalling problem. When modelled as graph problems, (a) is the most general case while (b), (c), (d) are restrictions to different graph classes. We show that (a) does not admit polynomial-time algorithms (assuming $${{\,\mathrm{\texttt {P}}\,}}\ne {{\,\mathrm{\texttt {NP}}\,}}$$ P ≠ NP ), even for relaxed variants of the problem (assuming $${{\,\mathrm{\texttt {P}}\,}}\ne {{\,\mathrm{\texttt {PSPACE}}\,}}$$ P ≠ PSPACE ). For (b), (c), (d), we present polynomial-time algorithms to solve the respective problems. We also generalize the problem to when at most k (for a fixed integer $$k\ge 2$$ k ≥ 2 ) contiguous vertices on a shortest path can be changed at a time. Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) repaving road networks, (b) rerouting data packets in a synchronous multiprocessing setting, (c) the shipping container stowage problem, and (d) the train marshalling problem. When modelled as graph problems, (a) is the most general case while (b), (c), (d) are restrictions to different graph classes. We show that (a) does not admit polynomial-time algorithms (assuming ), even for relaxed variants of the problem (assuming ). For (b), (c), (d), we present polynomial-time algorithms to solve the respective problems. We also generalize the problem to when at most (for a fixed integer ) contiguous vertices on a shortest path can be changed at a time.
Author	Jha, Agastya Vibhuti Lahiri, Abhiruk Gajjar, Kshitij Kumar, Manish
Author_xml	– sequence: 1 givenname: Kshitij surname: Gajjar fullname: Gajjar, Kshitij organization: Indian Institute of Technology Jodhpur – sequence: 2 givenname: Agastya Vibhuti surname: Jha fullname: Jha, Agastya Vibhuti organization: École polytechnique fédérale de Lausanne – sequence: 3 givenname: Manish surname: Kumar fullname: Kumar, Manish organization: Ben-Gurion University of the Negev & Bar-ilan University – sequence: 4 givenname: Abhiruk surname: Lahiri fullname: Lahiri, Abhiruk email: abhiruk@iuuk.mff.cuni.cz organization: Charles University
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Keywords	Bridged graph Line graph Hardness of approximation Shortest path PSPACE-complete Circle graph Reconfiguration Boolean hypercube
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Snippet	Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time so that all the intermediate paths...
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Apexes Cargo containers Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Multiprocessing Packets (communication) Polynomials Roads & highways Shortest-path problems Stowage (onboard equipment) Theory of Computation
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Title	Reconfiguring Shortest Paths in Graphs
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