The Parameterised Complexity Of Integer Multicommodity Flow

Authors Hans L. Bodlaender , Isja Mannens , Jelle J. Oostveen , Sukanya Pandey , Erik Jan van Leeuwen



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Author Details

Hans L. Bodlaender
  • Utrecht University, The Netherlands
Isja Mannens
  • Utrecht University, The Netherlands
Jelle J. Oostveen
  • Utrecht University, The Netherlands
Sukanya Pandey
  • Utrecht University, The Netherlands
Erik Jan van Leeuwen
  • Utrecht University, The Netherlands

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Hans L. Bodlaender, Isja Mannens, Jelle J. Oostveen, Sukanya Pandey, and Erik Jan van Leeuwen. The Parameterised Complexity Of Integer Multicommodity Flow. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.IPEC.2023.6

Abstract

The Integer Multicommodity Flow problem has been studied extensively in the literature. However, from a parameterised perspective, mostly special cases, such as the Disjoint Path problem, have been considered. Therefore, we investigate the parameterised complexity of the general Integer Multicommodity Flow problem. We show that the decision version of this problem on directed graphs for a constant number of commodities, when the capacities are given in unary, is XNLP-complete with pathwidth as parameter and XALP-complete with treewidth as parameter. When the capacities are given in binary, the problem is NP-complete even for graphs of pathwidth at most 13. We give related results for undirected graphs. These results imply that the problem is unlikely to be fixed-parameter tractable by these parameters. In contrast, we show that the problem does become fixed-parameter tractable when weighted tree partition width (a variant of tree partition width for edge weighted graphs) is used as parameter.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Problems, reductions and completeness
Keywords
  • multicommodity flow
  • parameterised complexity
  • XNLP-completeness
  • XALP-completeness

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References

  1. Cynthia Barnhart, Niranjan Krishnan, and Pamela H. Vance. Multicommodity flow problems. In Christodoulos A. Floudas and Panos M. Pardalos, editors, Encyclopedia of Optimization, pages 2354-2362. Springer US, 2009. URL: https://doi.org/10.1007/978-0-387-74759-0_407.
  2. Hans L. Bodlaender, Gunther Cornelissen, and Marieke van der Wegen. Problems hard for treewidth but easy for stable gonality. In Michael A. Bekos and Michael Kaufmann, editors, Proceedings 48th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2022, volume 13453 of Lecture Notes in Computer Science, pages 84-97. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-15914-5_7.
  3. Hans L. Bodlaender, Gunther Cornelissen, and Marieke van der Wegen. Problems hard for treewidth but easy for stable gonality. arXiv, abs/2202.06838, 2022. URL: https://arxiv.org/abs/2202.06838.
  4. Hans L. Bodlaender, Carla Groenland, and Hugo Jacob. On the parameterized complexity of computing tree-partitions. In Holger Dell and Jesper Nederlof, editors, 17th International Symposium on Parameterized and Exact Computation, IPEC 2022, September 7-9, 2022, Potsdam, Germany, volume 249 of LIPIcs, pages 7:1-7:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.IPEC.2022.7.
  5. Hans L. Bodlaender, Carla Groenland, Hugo Jacob, Lars Jaffke, and Paloma T. Lima. XNLP-completeness for parameterized problems on graphs with a linear structure. In Holger Dell and Jesper Nederlof, editors, Proceedings 17th International Symposium on Parameterized and Exact Computation, IPEC 2022, volume 249 of LIPIcs, pages 8:1-8:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.IPEC.2022.8.
  6. Hans L. Bodlaender, Carla Groenland, Hugo Jacob, Marcin Pilipczuk, and Michal Pilipczuk. On the complexity of problems on tree-structured graphs. In Holger Dell and Jesper Nederlof, editors, Proceedings 17th International Symposium on Parameterized and Exact Computation, IPEC 2022, volume 249 of LIPIcs, pages 6:1-6:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.IPEC.2022.6.
  7. Hans L. Bodlaender, Carla Groenland, Jesper Nederlof, and Céline M. F. Swennenhuis. Parameterized problems complete for nondeterministic FPT time and logarithmic space. In Proceedings 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, pages 193-204. IEEE, 2022. URL: https://doi.org/10.1109/FOCS52979.2021.00027.
  8. Hans L. Bodlaender and Marieke van der Wegen. Parameterized complexity of scheduling chains of jobs with delays. In Yixin Cao and Marcin Pilipczuk, editors, 15th International Symposium on Parameterized and Exact Computation, IPEC 2020, volume 180 of LIPIcs, pages 4:1-4:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.IPEC.2020.4.
  9. Michael Elberfeld, Christoph Stockhusen, and Till Tantau. On the space and circuit complexity of parameterized problems: Classes and completeness. Algorithmica, 71(3):661-701, 2015. URL: https://doi.org/10.1007/s00453-014-9944-y.
  10. P. Erdös and P. Turán. On a problem of Sidon in additive number theory, and on some related problems. Journal of the London Mathematical Society, s1-16(4):212-215, 1941. URL: https://doi.org/10.1112/jlms/s1-16.4.212.
  11. Shimon Even, Alon Itai, and Adi Shamir. On the complexity of timetable and multicommodity flow problems. SIAM J. Comput., 5(4):691-703, 1976. URL: https://doi.org/10.1137/0205048.
  12. Michael R. Fellows, Danny Hermelin, Frances A. Rosamond, and Stéphane Vialette. On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci., 410(1):53-61, 2009. URL: https://doi.org/10.1016/j.tcs.2008.09.065.
  13. Krzysztof Fleszar, Matthias Mnich, and Joachim Spoerhase. New algorithms for maximum disjoint paths based on tree-likeness. Math. Program., 171(1-2):433-461, 2018. URL: https://doi.org/10.1007/s10107-017-1199-3.
  14. Steven Fortune, John E. Hopcroft, and James Wyllie. The directed subgraph homeomorphism problem. Theor. Comput. Sci., 10:111-121, 1980. URL: https://doi.org/10.1016/0304-3975(80)90009-2.
  15. A. Frank and Éva Tardos. An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica, 7(1):49-65, January 1987. URL: https://doi.org/10.1007/BF02579200.
  16. András Frank. Packing paths, circuits, and cuts - a survey. In Bernhard Korte, Lásló Lovász, Hans Jürgen Prömel, and Alexander Schrijver, editors, Paths, Flows, and VLSI-Layout, pages 47-100. Springer-Verlag, Berlin, 1990. Google Scholar
  17. Tobias Friedrich, Davis Issac, Nikhil Kumar, Nadym Mallek, and Ziena Zeif. Approximate max-flow min-multicut theorem for graphs of bounded treewidth. arXiv, abs/2211.06267, 2022. URL: https://arxiv.org/abs/2211.06267.
  18. Tobias Friedrich, Davis Issac, Nikhil Kumar, Nadym Mallek, and Ziena Zeif. A primal-dual algorithm for multicommodity flows and multicuts in treewidth-2 graphs. In Amit Chakrabarti and Chaitanya Swamy, editors, Proceedings 25th International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 26th International Conference on Randomization and Computation APPROX/RANDOM 2022, volume 245 of LIPIcs, pages 55:1-55:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.55.
  19. Robert Ganian and Sebastian Ordyniak. The power of cut-based parameters for computing edge-disjoint paths. Algorithmica, 83(2):726-752, 2021. URL: https://doi.org/10.1007/s00453-020-00772-w.
  20. Robert Ganian, Sebastian Ordyniak, and M. S. Ramanujan. On structural parameterizations of the edge disjoint paths problem. Algorithmica, 83(6):1605-1637, 2021. URL: https://doi.org/10.1007/s00453-020-00795-3.
  21. Venkatesan Guruswami, Sanjeev Khanna, Rajmohan Rajaraman, F. Bruce Shepherd, and Mihalis Yannakakis. Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems. Journal of Computing and System Sciences, 67(3):473-496, 2003. URL: https://doi.org/10.1016/S0022-0000(03)00066-7.
  22. George Karakostas. Faster approximation schemes for fractional multicommodity flow problems. ACM Trans. Algorithms, 4(1):13:1-13:17, 2008. URL: https://doi.org/10.1145/1328911.1328924.
  23. Richard M. Karp. Reducibility among combinatorial problems. In Raymond E. Miller and James W. Thatcher, editors, Proceedings of a symposium on the Complexity of Computer Computations, held March 20-22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, USA, The IBM Research Symposia Series, pages 85-103. Plenum Press, New York, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  24. Ken-ichi Kawarabayashi, Yusuke Kobayashi, and Bruce A. Reed. The disjoint paths problem in quadratic time. J. Comb. Theory, Ser. B, 102(2):424-435, 2012. URL: https://doi.org/10.1016/j.jctb.2011.07.004.
  25. Bernhard Korte and Jens Vygen. Combinatorial Optimization: Theory and Algorithms. Springer Berlin, Heidelberg, 2000. Google Scholar
  26. Takao Nishizeki, Jens Vygen, and Xiao Zhou. The edge-disjoint paths problem is np-complete for series-parallel graphs. Discret. Appl. Math., 115(1-3):177-186, 2001. URL: https://doi.org/10.1016/S0166-218X(01)00223-2.
  27. Michal Pilipczuk and Marcin Wrochna. On space efficiency of algorithms working on structural decompositions of graphs. ACM Trans. Comput. Theory, 9(4):18:1-18:36, 2018. URL: https://doi.org/10.1145/3154856.
  28. Neil Robertson and Paul D. Seymour. Graph minors .xiii. the disjoint paths problem. J. Comb. Theory, Ser. B, 63(1):65-110, 1995. URL: https://doi.org/10.1006/jctb.1995.1006.
  29. Detlef Seese. Tree-partite graphs and the complexity of algorithms. In Lothar Budach, editor, 5th International Conference on Fundamentals of Computation Theory, FCT 1985, volume 199 of Lecture Notes in Computer Science, pages 412-421. Springer, 1985. URL: https://doi.org/10.1007/BFb0028825.
  30. F. Bruce Shepherd and Adrian R. Vetta. The inapproximability of maximum single-sink unsplittable, priority and confluent flow problems. Theory of Computation, 13(1):1-25, 2017. URL: https://doi.org/10.4086/toc.2017.v013a020.
  31. Jens Vygen. Disjoint paths. Technical Report 94816, Research Institute for Discrete Mathematics, University of Bonn, 1998. Google Scholar
  32. I-Lin Wang. Multicommodity network flows: A survey, part I: Applications and formulations. International Journal of Operations Research, 15(4):145-153, 2018. URL: http://www.orstw.org.tw/ijor/vol15no4/IJOR2018_vol15_no4_p145_p153.pdf.
  33. I-Lin Wang. Multicommodity network flows: A survey, part II: Solution methods. International Journal of Operations Research, 15(4):155-173, 2018. URL: http://www.orstw.org.tw/ijor/vol15no4/IJOR2018_vol15_no4_p155_p173.pdf.
  34. Xiao Zhou, Syurei Tamura, and Takao Nishizeki. Finding edge-disjoint paths in partial k-trees. Algorithmica, 26(1):3-30, 2000. URL: https://doi.org/10.1007/s004539910002.