Eulerian Walks in Temporal Graphs
An Eulerian walk (or Eulerian trail) is a walk (resp. trail) that visits every edge of a graph G at least (resp. exactly) once. This notion was first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. But what if Euler had to take a bus? In a temporal g...
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01.03.2023
Springer Nature B.V |
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| Abstract | An Eulerian walk (or Eulerian trail) is a walk (resp. trail) that visits every edge of a graph
G
at least (resp. exactly) once. This notion was first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. But what if Euler had to take a bus? In a temporal graph
(
G
,
λ
)
, with
λ
:
E
(
G
)
→
2
[
τ
]
, an edge
e
∈
E
(
G
)
is available only at the times specified by
λ
(
e
)
⊆
[
τ
]
, in the same way the connections of the public transportation network of a city or of sightseeing tours are available only at scheduled times. In this paper, we deal with temporal walks, local trails, and trails, respectively referring to edge traversal with no constraints, constrained to not repeating the same edge in a single timestamp, and constrained to never repeating the same edge throughout the entire traversal. We show that, if the edges are always available, then deciding whether
(
G
,
λ
)
has a temporal walk or trail is polynomial, while deciding whether it has a local trail is
NP
-complete even if
τ
=
2
. In contrast, in the general case, solving any of these problems is
NP
-complete, even under very strict hypotheses. We finally give
XP
algorithms parametrized by
τ
for walks, and by
τ
+
t
w
(
G
)
for trails and local trails, where
t
w
(
G
)
refers to the treewidth of
G
. |
|---|---|
| AbstractList | An Eulerian walk (or Eulerian trail) is a walk (resp. trail) that visits every edge of a graph G at least (resp. exactly) once. This notion was first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. But what if Euler had to take a bus? In a temporal graph (G,λ), with λ:E(G)→2[τ], an edge e∈E(G) is available only at the times specified by λ(e)⊆[τ], in the same way the connections of the public transportation network of a city or of sightseeing tours are available only at scheduled times. In this paper, we deal with temporal walks, local trails, and trails, respectively referring to edge traversal with no constraints, constrained to not repeating the same edge in a single timestamp, and constrained to never repeating the same edge throughout the entire traversal. We show that, if the edges are always available, then deciding whether (G,λ) has a temporal walk or trail is polynomial, while deciding whether it has a local trail is NP-complete even if τ=2. In contrast, in the general case, solving any of these problems is NP-complete, even under very strict hypotheses. We finally give XP algorithms parametrized by τ for walks, and by τ+tw(G) for trails and local trails, where tw(G) refers to the treewidth of G. An Eulerian walk (or Eulerian trail) is a walk (resp. trail) that visits every edge of a graph G at least (resp. exactly) once. This notion was first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. But what if Euler had to take a bus? In a temporal graph ( G , λ ) , with λ : E ( G ) → 2 [ τ ] , an edge e ∈ E ( G ) is available only at the times specified by λ ( e ) ⊆ [ τ ] , in the same way the connections of the public transportation network of a city or of sightseeing tours are available only at scheduled times. In this paper, we deal with temporal walks, local trails, and trails, respectively referring to edge traversal with no constraints, constrained to not repeating the same edge in a single timestamp, and constrained to never repeating the same edge throughout the entire traversal. We show that, if the edges are always available, then deciding whether ( G , λ ) has a temporal walk or trail is polynomial, while deciding whether it has a local trail is NP -complete even if τ = 2 . In contrast, in the general case, solving any of these problems is NP -complete, even under very strict hypotheses. We finally give XP algorithms parametrized by τ for walks, and by τ + t w ( G ) for trails and local trails, where t w ( G ) refers to the treewidth of G . An Eulerian walk (or Eulerian trail) is a walk (resp. trail) that visits every edge of a graph G at least (resp. exactly) once. This notion was first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. But what if Euler had to take a bus? In a temporal graph $$\varvec{(G,\lambda )}$$ ( G , λ ) , with $$\varvec{\lambda : E(G)}\varvec{\rightarrow } \varvec{2}^{\varvec{[\tau ]}}$$ λ : E ( G ) → 2 [ τ ] , an edge $$\varvec{e}\varvec{\in } \varvec{E(G)}$$ e ∈ E ( G ) is available only at the times specified by $$\varvec{\lambda (e)}\varvec{\subseteq } \varvec{[\tau ]}$$ λ ( e ) ⊆ [ τ ] , in the same way the connections of the public transportation network of a city or of sightseeing tours are available only at scheduled times. In this paper, we deal with temporal walks, local trails, and trails, respectively referring to edge traversal with no constraints, constrained to not repeating the same edge in a single timestamp, and constrained to never repeating the same edge throughout the entire traversal. We show that, if the edges are always available, then deciding whether $$\varvec{(G,\lambda )}$$ ( G , λ ) has a temporal walk or trail is polynomial, while deciding whether it has a local trail is $$\varvec{\texttt {NP}}$$ NP -complete even if $$\varvec{\tau = 2}$$ τ = 2 . In contrast, in the general case, solving any of these problems is $$\varvec{\texttt {NP}}$$ NP -complete, even under very strict hypotheses. We finally give $$\varvec{\texttt {XP}}$$ XP algorithms parametrized by $$\varvec{\tau }$$ τ for walks, and by $$\varvec{\tau +tw(G)}$$ τ + t w ( G ) for trails and local trails, where $$\varvec{tw(G)}$$ t w ( G ) refers to the treewidth of $$\varvec{G}$$ G . |
| Author | Marino, Andrea Silva, Ana |
| Author_xml | – sequence: 1 givenname: Andrea orcidid: 0000-0002-9854-7885 surname: Marino fullname: Marino, Andrea email: andrea.marino@unifi.it organization: Dipartimento di Statistica, Informatica, Applicazioni, Università degli Studi di Firenze – sequence: 2 givenname: Ana orcidid: 0000-0001-8917-0564 surname: Silva fullname: Silva, Ana organization: Dipartimento di Statistica, Informatica, Applicazioni, Università degli Studi di Firenze, Departamento de Matemática, Universidade Federal do Ceará |
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| Cites_doi | 10.1007/s10100-018-0598-8 10.1080/15427951.2016.1177801 10.1137/130947374 10.1155/2013/721051 10.1007/s00453-012-9667-x 10.1007/BFb0045375 10.1007/978-3-662-47672-7_36 10.1007/978-3-030-00256-5_19 10.1007/978-3-030-54921-3_8 10.1137/0205049 10.1016/B978-0-12-566780-7.50022-2 10.1007/978-3-030-17402-6_2 10.1080/17445760.2012.668546 10.1137/S0097539792225297 10.1016/j.tcs.2016.04.006 10.1007/s13278-018-0537-7 10.1016/S0898-1221(02)00156-6 10.1145/335305.335364 10.1007/978-3-030-79987-8_34 10.1002/net.3230170304 10.1007/978-3-030-79987-8_8 10.1137/130936816 |
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| Keywords | Temporal graphs Eulerian trails Bounded treewidth Edge cover Parameterized complexity Eulerian walks |
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| References | DrorMSternHTrudeauPPostman tour on a graph with precedence relation on arcsNetworks198717328329489816810.1002/net.32301703040644.90090 BodlaenderHLDrangePGDregiMSA ckn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c^k n$$\end{document} 5-approximation algorithm for treewidthSIAM J. Comput.2016452317378347970510.1137/1309473741333.05282 Euler, L.: Solutio problematis ad geometriam situs pertinentis. Commentarii academiae scientiarum Petropolitanae pp. 128–140 (1741) FominFVGolovachPALong circuits and large Euler subgraphsSIAM J. Discret. Math.2014282878892321379310.1137/1309368161298.05300 Kempe, D., Kleinberg, J.M., Kumar, A.: Connectivity and inference problems for temporal networks. In: Yao FF, Luks EM (eds) Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, May 21–23, pp. 504–513 . ACM, Portland (2000) Kloks, T.: Treewidth, Computations and Approximations. Lecture Notes in Computer Science, vol. 842. Springer (1994) LatapyM ViardTMagnienCStream graphs and link streams for the modeling of interactions over timeSoc. Netw. Anal. Min.20188161:161:2910.1007/s13278-018-0537-71426.68227 Marino, A., Silva, A.: Königsberg sightseeing: Eulerian walks in temporal graphs. In: Flocchini, P., Moura, L. (eds) Combinatorial Algorithms - 32nd International Workshop, IWOCA 2021, Ottawa, ON, Canada, July 5–7, 2021, Proceedings. Lecture Notes in Computer Science, vol. 12757, pp. 485–500. Springer (2021) DahlhausEJohnsonDSPapadimitriouCHThe complexity of multiterminal cutsSIAM J. Comput.1994234864894128357910.1137/S00975397922252970809.68075 MichailOSpirakisPGTraveling salesman problems in temporal graphsTheoret. Comput. Sci.2016634123350127410.1016/j.tcs.2016.04.0061338.90349 Manuel, P.: Revisiting path-type covering and partitioning problems. arXiv:1807.10613 (2018) WangHFWenYPTime-constrained Chinese postman problemsComput. Math. Appl.2002443–4375387191283610.1016/S0898-1221(02)00156-61067.90146 CyganMMarxDPilipczukMParameterized complexity of Eulerian deletion problemsAlgorithmica20146814161314747510.1007/s00453-012-9667-x1284.05157 Erlebach, T., Spooner, J.T.: Faster exploration of degree-bounded temporal graphs. In: Potapov I, Spirakis PG, Worrell J (eds) 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018, August 27–31, 2018, Liverpool, UK, LIPIcs, vol 117. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, pp. 36:1–36:13 (2018) GuanMGraphic programming using odd or even points. Acta Mathematica Sinica (in Chinese) 10:263–266. Translated in Chinese Mathematics 1Am. Math. 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| References_xml | – reference: FominFVGolovachPALong circuits and large Euler subgraphsSIAM J. Discret. Math.2014282878892321379310.1137/1309368161298.05300 – reference: MichailOSpirakisPGTraveling salesman problems in temporal graphsTheoret. Comput. Sci.2016634123350127410.1016/j.tcs.2016.04.0061338.90349 – reference: Erlebach, T., Spooner, J.T.: Non-strict temporal exploration. In: Richa, A.W., Scheideler, C. (eds) Structural Information and Communication Complexity—27th International Colloquium, SIROCCO 2020, Paderborn, Germany, June 29–July 1, 2020, Proceedings. Lecture Notes in Computer Science, vol. 12156, pp. 129–145. Springer (2020) – reference: CasteigtsAFlocchiniPQuattrociocchiWTime-varying graphs and dynamic networksInt. J. Parallel Emerg. Distrib. Syst.201227538740810.1080/17445760.2012.668546 – reference: DrorMSternHTrudeauPPostman tour on a graph with precedence relation on arcsNetworks198717328329489816810.1002/net.32301703040644.90090 – reference: WestDIntroduction to Graph Theory1996Upper Saddle RiverPrentice Hall0845.05001 – reference: ÇodurMKYılmazMA time-dependent hierarchical Chinese postman problemCEJOR2020281337366405013610.1007/s10100-018-0598-807176691 – reference: Orlin, J.B.: Some problems on dynamic/periodic graphs. In: Progress in Combinatorial Optimization, pp. 273–293. Elsevier (1984) – reference: GareyMRJohnsonDSTarjanREThe planar Hamiltonian circuit problem is NP-completeSIAM J. Comput.19765470471444451610.1137/02050490346.05110 – reference: Erlebach, T., Hoffmann, M., Kammer, F.: On temporal graph exploration. In: 42nd International Colloquium on Automata, Languages, and Programming—ICALP 2015, Kyoto, Japan, Lecture Notes in Computer Science, vol. 9134, pp. 444–455. Springer (2015) – reference: LatapyM ViardTMagnienCStream graphs and link streams for the modeling of interactions over timeSoc. Netw. Anal. Min.20188161:161:2910.1007/s13278-018-0537-71426.68227 – reference: ArumugamSHamidIAbrahamVDecomposition of graphs into paths and cyclesJ. Discrete Math.201320131610.1155/2013/7210511295.05184 – reference: Borgnat, P., Fleury, E., Guillaume, J., et al.: Evolving networks. In: Mining Massive Data Sets for Security, pp. 198–203 (2007) – reference: CyganMMarxDPilipczukMParameterized complexity of Eulerian deletion problemsAlgorithmica20146814161314747510.1007/s00453-012-9667-x1284.05157 – reference: Kloks, T.: Treewidth, Computations and Approximations. Lecture Notes in Computer Science, vol. 842. 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| Snippet | An Eulerian walk (or Eulerian trail) is a walk (resp. trail) that visits every edge of a graph
G
at least (resp. exactly) once. This notion was first discussed... An Eulerian walk (or Eulerian trail) is a walk (resp. trail) that visits every edge of a graph G at least (resp. exactly) once. This notion was first discussed... |
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| SubjectTerms | Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Constraints Data Structures and Information Theory Mathematics of Computing Polynomials Public transportation Special Issue on Combinatorial Algorithms (IWOCA 2021) Theory of Computation Transportation networks |
| Title | Eulerian Walks in Temporal Graphs |
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