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Edge-Minimum Walk of Modular Length in Polynomial Time

Authors Antoine Amarilli , Benoît Groz , Nicole Wein

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Author Details

Antoine Amarilli
  • Univ. Lille, Inria, CNRS, Centrale Lille, UMR 9189 CRIStAL, F-59000 Lille, France
  • LTCI, Télécom Paris, Institut polytechnique de Paris, France
Benoît Groz
  • Paris-Saclay University, CNRS, LISN, France
Nicole Wein
  • University of Michigan, Ann Arbor, MI, USA

Funding

  • Amarilli, Antoine: Amarilli was partially supported by the ANR project EQUUS ANR-19-CE48-0019, by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 431183758, and by the ANR project ANR-18-CE23-0003-02 (“CQFD”).

Acknowledgements

We are grateful to the reviewers of the conference version for their helpful feedback. We also would like to thank the Simons Institute Fall 2023 programs "Logic and Algorithms in Database Theory and AI" and "Data Structures and Optimization for Fast Algorithms" for the initiation of this work. Last, we are grateful to Xiao Hu and Mikaël Monet for early discussions.

Cite As Get BibTex

Antoine Amarilli, Benoît Groz, and Nicole Wein. Edge-Minimum Walk of Modular Length in Polynomial Time. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 5:1-5:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.5

Abstract

We study the problem of finding, in a directed graph, an st-walk of length r od q which is edge-minimum, i.e., uses the smallest number of distinct edges. Despite the vast literature on paths and cycles with modularity constraints, to the best of our knowledge we are the first to study this problem. Our main result is a polynomial-time algorithm that solves this task when r and q are constants. 
We also show how our proof technique gives an algorithm to solve a generalization of the well-known Directed Steiner Network problem, in which connections between endpoint pairs are required to satisfy modularity constraints on their length. Our algorithm is polynomial when the number of endpoint pairs and the modularity constraints on the pairs are constants.
In this version of the article, proofs and examples are omitted because of space constraints. Detailed proofs are available in the full version [Antoine Amarilli et al., 2024].

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Shortest paths
Keywords
  • Directed Steiner Network
  • Modularity

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References

  1. Noga Alon and Michael Krivelevich. Divisible subdivisions. J. Graph Theory, 98(4), 2021. URL: https://doi.org/10.1002/jgt.22716.
  2. Antoine Amarilli. Survey of results on the ModPath and ModCycle problems, 2024. URL: https://doi.org/10.48550/arXiv.2409.00770.
  3. Antoine Amarilli, Benoît Groz, and Nicole Wein. Edge-minimum walk of modular length in polynomial time, 2024. URL: https://arxiv.org/abs/2412.01614.
  4. Esther M. Arkin, Christos H. Papadimitriou, and Mihalis Yannakakis. Modularity of cycles and paths in graphs. J. ACM, 38(2), 1991. URL: https://doi.org/10.1145/103516.103517.
  5. Guillaume Bagan, Angela Bonifati, and Benoît Groz. A trichotomy for regular simple path queries on graphs. J. Comput. Syst. Sci., 108, 2020. URL: https://doi.org/10.1016/J.JCSS.2019.08.006.
  6. Andreas Björklund, Thore Husfeldt, and Petteri Kaski. The shortest even cycle problem is tractable. In STOC, 2022. URL: https://doi.org/10.1145/3519935.3520030.
  7. Archit Chauhan, Samir Datta, Chetan Gupta, and Vimal Raj Sharma. The even-path problem in directed single-crossing-minor-free graphs, 2024. URL: https://doi.org/10.48550/arXiv.2407.00237.
  8. Chandra Chekuri, Guy Even, Anupam Gupta, and Danny Segev. Set connectivity problems in undirected graphs and the directed Steiner network problem. ACM Trans. Algorithms, 7(2), 2011. URL: https://doi.org/10.1145/1921659.1921664.
  9. Rajesh Chitnis, Andreas Emil Feldmann, and Pasin Manurangsi. Parameterized approximation algorithms for bidirected Steiner network problems. ACM Trans. Algorithms, 17(2), 2021. URL: https://doi.org/10.1145/3447584.
  10. Fan R. K. Chung, Wayne Goddard, and Daniel J. Kleitman. Even cycles in directed graphs. SIAM J. Discret. Math., 7(3), 1994. URL: https://doi.org/10.1137/S0895480192225433.
  11. Mariano P. Consens and Alberto O. Mendelzon. Graphlog: a visual formalism for real life recursion. In PODS, 1990. URL: https://doi.org/10.1145/298514.298591.
  12. Isabel F. Cruz, Alberto O. Mendelzon, and Peter T. Wood. A graphical query language supporting recursion. In SIGMOD, 1987. URL: https://doi.org/10.1145/38713.38749.
  13. Gianlorenzo D'Angelo and Esmaeil Delfaraz. Approximation algorithms for node-weighted directed Steiner problems. In IWOCA, 2024. URL: https://doi.org/10.1007/978-3-031-63021-7_21.
  14. Shagnik Das, Nemanja Draganić, and Raphael Steiner. Tight bounds for divisible subdivisions. J. Combin. Theory Ser. B, 165, 2024. URL: https://doi.org/10.1016/j.jctb.2023.10.011.
  15. Irit Dinur and Pasin Manurangsi. ETH-hardness of approximating 2-CSPs and directed Steiner network. In ITCS, 2018. URL: https://doi.org/10.4230/LIPICS.ITCS.2018.36.
  16. Ajit A Diwan. Cycles of weight divisible by k, 2024. URL: https://doi.org/10.48550/arXiv.2407.01198.
  17. Jon Feldman and Matthias Ruhl. The directed Steiner network problem is tractable for a constant number of terminals. SIAM J. Comput., 36(2), 2006. URL: https://doi.org/10.1137/S0097539704441241.
  18. Andreas Emil Feldmann and Dániel Marx. The complexity landscape of fixed-parameter directed Steiner network problems. ACM Trans. Comput. Theory, 15(3–4), 2023. URL: https://doi.org/10.1145/3580376.
  19. Steven Fortune, John E. Hopcroft, and James Wyllie. The directed subgraph homeomorphism problem. Theor. Comput. Sci., 10, 1980. URL: https://doi.org/10.1016/0304-3975(80)90009-2.
  20. Esther Galby, Sándor Kisfaludi-Bak, Dániel Marx, and Roohani Sharma. Subexponential parameterized directed Steiner network problems on planar graphs: A complete classification. In ICALP, 2024. URL: https://doi.org/10.4230/LIPICS.ICALP.2024.67.
  21. Jiong Guo, Rolf Niedermeier, and Ondrej Suchý. Parameterized complexity of arc-weighted directed Steiner problems. SIAM J. Discret. Math., 25(2), 2011. URL: https://doi.org/10.1137/100794560.
  22. Xiao Hu and Stavros Sintos. Finding smallest witnesses for conjunctive queries. In ICDT, 2024. URL: https://doi.org/10.4230/LIPICS.ICDT.2024.24.
  23. Alpár Jüttner, Csaba Király, Lydia Mirabel Mendoza-Cadena, Gyula Pap, Ildikó Schlotter, and Yutaro Yamaguchi. Shortest odd paths in undirected graphs with conservative weight functions. Discrete Appl. Math., 357, 2024. URL: https://doi.org/10.1016/j.dam.2024.05.044.
  24. Naonori Kakimura and Ken-ichi Kawarabayashi. Packing cycles through prescribed vertices under modularity constraints. Adv. in Appl. Math., 49(2), 2012. URL: https://doi.org/10.1016/j.aam.2012.03.002.
  25. Ken-ichi Kawarabayashi, Stephan Kreutzer, O-joung Kwon, and Qiqin Xie. A half-integral Erdos-Posa theorem for directed odd cycles. In SODA, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch118.
  26. Andrea S. LaPaugh and Christos H. Papadimitriou. The even-path problem for graphs and digraphs. Networks, 14(4), 1984. URL: https://doi.org/10.1002/NET.3230140403.
  27. Anna Lubiw. A note on odd/even cycles. Discret. Appl. Math., 22(1), 1988. URL: https://doi.org/10.1016/0166-218X(88)90125-4.
  28. Wim Martens, Matthias Niewerth, and Tina Popp. A trichotomy for regular trail queries. Log. Methods Comput. Sci., 19(4), 2023. URL: https://doi.org/10.46298/LMCS-19(4:20)2023.
  29. William McCuaig. Pólya’s permanent problem. Electron. J. Comb., 11(1), 2004. URL: https://doi.org/10.37236/1832.
  30. Zhengjie Miao, Sudeepa Roy, and Jun Yang. Explaining wrong queries using small examples. In SIGMOD, 2019. URL: https://doi.org/10.1145/3299869.3319866.
  31. Burkhard Monien. The complexity of determining a shortest cycle of even length. Computing, 31(4), 1983. URL: https://doi.org/10.1007/BF02251238.
  32. Zhivko Prodanov Nedev. Finding an even simple path in a directed planar graph. SIAM J. Comput., 29(2), 1999. URL: https://doi.org/10.1137/S0097539797330343.
  33. Mehdy Roayaei and Mohammadreza Razzazi. Parameterized complexity of directed Steiner network with respect to shared vertices and arcs. Int. J. Found. Comput. Sci., 29(7), 2018. URL: https://doi.org/10.1142/S0129054118500302.
  34. Neil Robertson, P. D. Seymour, and Robin Thomas. Permanents, Pfaffian orientations, and even directed circuits. Ann. of Math. (2), 150(3), 1999. URL: https://doi.org/10.2307/121059.
  35. Ildikó Schlotter and András Sebő. Odd paths, cycles and t-joins: Connections and algorithms, 2022. URL: https://arxiv.org/abs/2211.12862.
  36. Paul Seymour and Carsten Thomassen. Characterization of even directed graphs. J. Combin. Theory Ser. B, 42(1), 1987. URL: https://doi.org/10.1016/0095-8956(87)90061-X.
  37. Carsten Thomassen. Even cycles in directed graphs. Eur. J. Comb., 6(1), 1985. URL: https://doi.org/10.1016/S0195-6698(85)80025-1.
  38. Carsten Thomassen. The even cycle problem for planar digraphs. J. Algorithms, 15(1), 1993. URL: https://doi.org/10.1006/jagm.1993.1030.
  39. Vijay V. Vazirani and Mihalis Yannakakis. Pfaffian orientations, 0-1 permanents, and even cycles in directed graphs. Discret. Appl. Math., 25(1-2), 1989. URL: https://doi.org/10.1016/0166-218X(89)90053-X.
  40. Paul Wollan. Packing cycles with modularity constraints. Combinatorica, 31(1), 2011. URL: https://doi.org/10.1007/s00493-011-2551-5.
  41. Raphael Yuster and Uri Zwick. Finding even cycles even faster. SIAM J. Discrete Math., 10(2), 1997. URL: https://doi.org/10.1137/S0895480194274133.
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