Document Open Access Logo

Improved Algorithms for Computing the Cycle of Minimum Cost-to-Time Ratio in Directed Graphs

Authors Karl Bringmann, Thomas Dueholm Hansen, Sebastian Krinninger

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2017.124.pdf
  • Filesize: 0.53 MB
  • 16 pages

Document Identifiers

Subject Classification

Keywords
  • quantitative verification and synthesis
  • parametric search
  • shortest paths
  • negative cycle detection

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We study the problem of finding the cycle of minimum cost-to-time ratio in a directed graph with n nodes and m edges. This problem has a long history in combinatorial optimization and has recently seen interesting applications in the context of quantitative verification. We focus on strongly polynomial algorithms to cover the use-case where the weights are relatively large compared to the size of the graph. Our main result is an algorithm with running time ~O(m^{3/4} n^{3/2}), which gives the first improvement over Megiddo's ~O(n^3) algorithm [JACM'83] for sparse graphs (We use the notation ~O(.) to hide factors that are polylogarithmic in n.) We further demonstrate how to obtain both an algorithm with running time n^3/2^{Omega(sqrt(log n)} on general graphs and an algorithm with running time ~O(n) on constant treewidth graphs. To obtain our main result, we develop a parallel algorithm for negative cycle detection and single-source shortest paths that might be of independent interest.

Cite As Get BibTex

Karl Bringmann, Thomas Dueholm Hansen, and Sebastian Krinninger. Improved Algorithms for Computing the Cycle of Minimum Cost-to-Time Ratio in Directed Graphs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 124:1-124:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.ICALP.2017.124

Author Details

Karl Bringmann
Thomas Dueholm Hansen
Sebastian Krinninger

References

  1. Pankaj K. Agarwal, Micha Sharir, and Sivan Toledo. Applications of parametric searching in geometric optimization. Journal of Algorithms, 17(3):292-318, 1994. Announced at SODA'92. URL: http://dx.doi.org/10.1006/jagm.1994.1038.
  2. Giorgio Ausiello, Alessandro D'Atri, and Marco Protasi. Structure preserving reductions among convex optimization problems. Journal of Computer and System Sciences, 21(1):136-153, 1980. Announced at ICALP'77. URL: http://dx.doi.org/10.1016/0022-0000(80)90046-X.
  3. Richard Bellman. On a routing problem. Quarterly of Applied Mathematics, 16(1):87-90, 1958. Google Scholar
  4. Bonnie Berger, John Rompel, and Peter W. Shor. Efficient NC algorithms for set cover with applications to learning and geometry. Journal of Computer and System Sciences, 49(3):454-477, 1994. Announced at FOCS'89. URL: http://dx.doi.org/10.1016/S0022-0000(05)80068-6.
  5. Guy E. Blelloch, Yan Gu, Yihan Sun, and Kanat Tangwongsan. Parallel shortest paths using radius stepping. In Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 443-454, 2016. URL: http://dx.doi.org/10.1145/2935764.2935765.
  6. Roderick Bloem, Krishnendu Chatterjee, Karin Greimel, Thomas A. Henzinger, Georg Hofferek, Barbara Jobstmann, Bettina Könighofer, and Robert Könighofer. Synthesizing robust systems. Acta Informatica, 51(3-4):193-220, 2014. Announced at FMCAD'09. URL: http://dx.doi.org/10.1007/s00236-013-0191-5.
  7. Roderick Bloem, Krishnendu Chatterjee, Thomas A. Henzinger, and Barbara Jobstmann. Better quality in synthesis through quantitative objectives. In International Conference on Computer-Aided Verification (CAV), pages 140-156, 2009. URL: http://dx.doi.org/10.1007/978-3-642-02658-4_14.
  8. Gerth Stølting Brodal, Jesper Larsson Träff, and Christos D. Zaroliagis. A parallel priority queue with constant time operations. Journal of Parallel and Distributed Computing, 49(1):4-21, 1998. Announced at IPPS'97. URL: http://dx.doi.org/10.1006/jpdc.1998.1425.
  9. Steven M. Burns. Performance Analysis and Optimization of Asynchronous Circuits. PhD thesis, California Institute of Technology, 1991. Published as technical report CS-TR-91-01. Google Scholar
  10. Pavol Cerný, Krishnendu Chatterjee, Thomas A. Henzinger, Arjun Radhakrishna, and Rohit Singh. Quantitative synthesis for concurrent programs. In International Conference on Computer-Aided Verification (CAV), pages 243-259, 2011. URL: http://dx.doi.org/10.1007/978-3-642-22110-1_20.
  11. Arindam Chakrabarti, Luca de Alfaro, Thomas A. Henzinger, and Mariëlle Stoelinga. Resource interfaces. In International Conference on Embedded Software (EMSOFT), pages 117-133, 2003. URL: http://dx.doi.org/10.1007/978-3-540-45212-6_9.
  12. Timothy M. Chan and Ryan Williams. Deterministic APSP, orthogonal vectors, and more: Quickly derandomizing Razborov-Smolensky. In Symposium on Discrete Algorithms (SODA), pages 1246-1255, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch87.
  13. Krishnendu Chatterjee, Laurent Doyen, and Thomas A. Henzinger. Quantitative languages. ACM Transactions on Computational Logic, 11(4):23:1-23:38, 2010. Announced at CSL'08. URL: http://dx.doi.org/10.1145/1805950.1805953.
  14. Krishnendu Chatterjee, Rasmus Ibsen-Jensen, and Andreas Pavlogiannis. Faster algorithms for quantitative verification in constant treewidth graphs. In International Conference on Computer-Aided Verification (CAV), pages 140-157, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21690-4_9.
  15. Krishnendu Chatterjee, Rasmus Ibsen-Jensen, and Andreas Pavlogiannis. Optimal reachability and a space-time tradeoff for distance queries in constant-treewidth graphs. In European Symposium on Algorithms (ESA), pages 28:1-28:17, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ESA.2016.28.
  16. Shiva Chaudhuri and Christos D. Zaroliagis. Shortest paths in digraphs of small treewdith. Part II: optimal parallel algorithms. Theoretical Computer Science, 203(2):205-223, 1998. Announced at ESA'95. URL: http://dx.doi.org/10.1016/S0304-3975(98)00021-8.
  17. Shiva Chaudhuri and Christos D. Zaroliagis. Shortest paths in digraphs of small treewidth. Part I: sequential algorithms. Algorithmica, 27(3):212-226, 2000. Announced at ICALP'95. URL: http://dx.doi.org/10.1007/s004530010016.
  18. Edith Cohen. Using selective path-doubling for parallel shortest-path computations. Journal of Algorithms, 22(1):30-56, 1997. Announced at ISTCS'93. URL: http://dx.doi.org/10.1006/jagm.1996.0813.
  19. Edith Cohen. Polylog-time and near-linear work approximation scheme for undirected shortest paths. Journal of the ACM, 47(1):132-166, 2000. Announced at STOC'94. URL: http://dx.doi.org/10.1145/331605.331610.
  20. Michael B. Cohen, Aleksander Madry, Piotr Sankowski, and Adrian Vladu. Negative-weight shortest paths and unit capacity minimum cost flow in Õ(m^10/7 logW) time. In Symposium on Discrete Algorithms (SODA), pages 752-771, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.48.
  21. Richard Cole. Slowing down sorting networks to obtain faster sorting algorithms. Journal of the ACM, 34(1):200-208, 1987. Announced at FOCS'84. URL: http://dx.doi.org/10.1145/7531.7537.
  22. G.B. Dantzig, W. Blattner, and M.R. Rao. Finding a cycle in a graph with minimum cost to time ratio with application to a ship routing problem. In P. Rosenstiehl, editor, Theory of Graphs, pages 77-84. Dunod, Paris and Gordon and Breach, New York, 1967. Google Scholar
  23. Ali Dasdan, Sandy Irani, and Rajesh K. Gupta. Efficient algorithms for optimum cycle mean and optimum cost to time ratio problems. In Design Automation Conference (DAC), pages 37-42, 1999. URL: http://dx.doi.org/10.1145/309847.309862.
  24. Manfred Droste, Werner Kuich, and Heiko Vogler, editors. Handbook of Weighted Automata. Springer, 2009. URL: http://dx.doi.org/10.1007/978-3-642-01492-5.
  25. Jack Edmonds and Richard M. Karp. Theoretical improvements in algorithmic efficiency for network flow problems. Journal of the ACM, 19(2):248-264, 1972. URL: http://dx.doi.org/10.1145/321694.321699.
  26. L. R. Ford. Network flow theory. Technical Report P-923, The RAND Corporation, 1956. Google Scholar
  27. Bennett Fox. Finding minimal cost-time ratio circuits. Operations Research, 17(3):546-551, 1969. Google Scholar
  28. Stephan Friedrichs and Christoph Lenzen. Parallel metric tree embedding based on an algebraic view on moore-bellman-ford. In Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 455-466, 2016. URL: http://dx.doi.org/10.1145/2935764.2935777.
  29. François Le Gall. Powers of tensors and fast matrix multiplication. In International Symposium on Symbolic and Algebraic Computation (ISSAC), pages 296-303, 2014. URL: http://dx.doi.org/10.1145/2608628.2608664.
  30. Sabih H. Gerez, Sonia M. Heemstra de Groot, and Otto E. Herrmann. A polynomial time algorithm for the computation of the iteration-period bound in recursive data flow graphs. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 39(1):49-52, 1992. URL: http://dx.doi.org/10.1109/81.109243.
  31. Andrew V. Goldberg. Scaling algorithms for the shortest paths problem. SIAM Journal on Computing, 24(3):494-504, 1995. Announced at SODA'93. URL: http://dx.doi.org/10.1137/S0097539792231179.
  32. Manfred v. Golitschek. Optimal cycles in doubly weighted graphs and approximation of bivariate functions by univariate ones. Numerische Mathematik, 39(1):65-84, 1982. Google Scholar
  33. Mark Hartmann and James B. Orlin. Finding minimum cost to time ratio cycles with small integral transit times. Networks, 23(6):567-574, 1993. URL: http://dx.doi.org/10.1002/net.3230230607.
  34. Alexander T. Ishii, Charles E. Leiserson, and Marios C. Papaefthymiou. An algorithm for the tramp steamer problem based on mean-weight cycles. Technical Report MIT/LCS/TM-457, Massachusetts Institute of Technology, 1991. Google Scholar
  35. Kazuhito Ito and Keshab K. Parhi. Determining the minimum iteration period of an algorithm. VLSI Signal Processing, 11(3):229-244, 1995. URL: http://dx.doi.org/10.1007/BF02107055.
  36. David S. Johnson. Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences, 9(3):256-278, 1974. Announced at STOC'73. URL: http://dx.doi.org/10.1016/S0022-0000(74)80044-9.
  37. Donald B. Johnson. Efficient algorithms for shortest paths in sparse networks. Journal of the ACM, 24(1):1-13, 1977. URL: http://dx.doi.org/10.1145/321992.321993.
  38. Richard M. Karp. A characterization of the minimum cycle mean in a digraph. Discrete Mathematics, 23(3):309-311, 1978. URL: http://dx.doi.org/10.1016/0012-365X(78)90011-0.
  39. Philip N. Klein and Sairam Subramanian. A randomized parallel algorithm for single-source shortest paths. Journal of Algorithms, 25(2):205-220, 1997. Announced at STOC'92. URL: http://dx.doi.org/10.1006/jagm.1997.0888.
  40. Eugene L. Lawler. Optimal cycles in doubly weighted linear graphs. In P. Rosenstiehl, editor, Theory of Graphs, pages 209-214. Dunod, Paris and Gordon and Breach, New York, 1967. Google Scholar
  41. Eugene L. Lawler. Combinatorial Optimization: Network and Matroids. Holt, Rinehart and Winston, New York, 1976. Google Scholar
  42. Nimrod Megiddo. Applying parallel computation algorithms in the design of serial algorithms. Journal of the ACM, 30(4):852-865, 1983. Announced at FOCS'81. URL: http://dx.doi.org/10.1145/2157.322410.
  43. Ulrich Meyer and Peter Sanders. Δ-stepping: a parallelizable shortest path algorithm. Journal of Algorithms, 49(1):114-152, 2003. Announced at ESA'98. URL: http://dx.doi.org/10.1016/S0196-6774(03)00076-2.
  44. Gary L. Miller, Richard Peng, Adrian Vladu, and Shen Chen Xu. Improved parallel algorithms for spanners and hopsets. In Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 192-201, 2015. URL: http://dx.doi.org/10.1145/2755573.2755574.
  45. E. F. Moore. The shortest path through a maze. In International Symposium on the Theory of Switching, pages 285-292, 1959. URL: http://dx.doi.org/10.1137/1.9781611974331.ch87.
  46. James B. Orlin. An O(nm) time algorithm for finding the min length directed cycle in a weighted graph. In Symposium on Discrete Algorithms (SODA), pages 1866-1879, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.122.
  47. Christos H. Papadimitriou. Efficient search for rationals. Information Processing Letters, 8(1):1-4, 1979. URL: http://dx.doi.org/10.1016/0020-0190(79)90079-6.
  48. Piotr Sankowski. Shortest paths in matrix multiplication time. In European Symposium on Algorithms (ESA), pages 770-778, 2005. URL: http://dx.doi.org/10.1007/11561071_68.
  49. Hanmao Shi and Thomas H. Spencer. Time-work tradeoffs of the single-source shortest paths problem. Journal of Algorithms, 30(1):19-32, 1999. URL: http://dx.doi.org/10.1006/jagm.1998.0968.
  50. Thomas H. Spencer. Time-work tradeoffs for parallel algorithms. Journal of the ACM, 44(5):742-778, 1997. Announced at SODA'91 and SPAA'91. URL: http://dx.doi.org/10.1145/265910.265923.
  51. Mikkel Thorup. Worst-case update times for fully-dynamic all-pairs shortest paths. In Symposium on Theory of Computing (STOC), pages 112-119, 2005. URL: http://dx.doi.org/10.1145/1060590.1060607.
  52. Jeffrey D. Ullman and Mihalis Yannakakis. High-probability parallel transitive-closure algorithms. SIAM Journal on Computing, 20(1):100-125, 1991. Announced at SPAA'90. URL: http://dx.doi.org/10.1137/0220006.
  53. Ryan Williams. Faster all-pairs shortest paths via circuit complexity. In Symposium on Theory of Computing (STOC), pages 664-673, 2014. URL: http://dx.doi.org/10.1145/2591796.2591811.
  54. Raphael Yuster and Uri Zwick. Answering distance queries in directed graphs using fast matrix multiplication. In Symposium on Foundations of Computer Science (FOCS), pages 389-396, 2005. URL: http://dx.doi.org/10.1109/SFCS.2005.20.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact