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Practical Minimum Path Cover

Authors Manuel Cáceres , Brendan Mumey , Santeri Toivonen, Alexandru I. Tomescu

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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Network flows
Keywords
  • minimum path cover
  • directed acyclic graph
  • maximum flow
  • parameterized algorithms
  • edge sparsification
  • algorithm engineering

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Abstract

Computing a minimum path cover (MPC) of a directed acyclic graph (DAG) is a fundamental problem with a myriad of applications, including reachability. Although it is known how to solve the problem by a simple reduction to minimum flow, recent theoretical advances exploit this idea to obtain algorithms parameterized by the number of paths of an MPC, known as the width. These results obtain fast [Mäkinen et al., TALG 2019] and even linear time [Cáceres et al., SODA 2022] algorithms in the small-width regime.
In this paper, we present the first publicly available high-performance implementation of state-of-the-art MPC algorithms, including the parameterized approaches. Our experiments on random DAGs show that parameterized algorithms are orders-of-magnitude faster on dense graphs. Additionally, we present new fast pre-processing heuristics based on transitive edge sparsification. We show that our heuristics improve MPC-solvers by orders of magnitude.

Cite As Get BibTex

Manuel Cáceres, Brendan Mumey, Santeri Toivonen, and Alexandru I. Tomescu. Practical Minimum Path Cover. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 3:1-3:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SEA.2024.3

Author Details

Manuel Cáceres
  • Department of Computer Science, University of Helsinki, Finland
  • Department of Computer Science, Aalto University, Finland
Brendan Mumey
  • School of Computer Science, Montana State University, Bozeman, MT, USA
Santeri Toivonen
  • Department of Computer Science, University of Helsinki, Finland
Alexandru I. Tomescu
  • Department of Computer Science, University of Helsinki, Finland

Funding

This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 851093, SAFEBIO), from the Research Council of Finland grants No. 322595, 352821, 346968, 358744, and has been supported by Helsinki Institute for Information Technology HIIT, and US National Science Foundation grants DBI-1759522 and DBI-2309902.

Supplementary Materials

References

  1. Ravindra K Ahujia, Thomas L Magnanti, and James B Orlin. Network flows: Theory, algorithms and applications. New Jersey: Prentice-Hall, 1993. Google Scholar
  2. Amnon B Barak and Paul Erdös. On the maximal number of strongly independent vertices in a random acyclic directed graph. SIAM Journal on Algebraic Discrete Methods, 5(4):508-514, 1984. Google Scholar
  3. Ruben Becker, Maximilian Fickert, and Andreas Karrenbauer. A novel dual ascent algorithm for solving the min-cost flow problem. In 2016 Proceedings of the Eighteenth Workshop on Algorithm Engineering and Experiments (ALENEX), pages 151-159. SIAM, 2016. Google Scholar
  4. Simone Bova, Robert Ganian, and Stefan Szeider. Model checking existential logic on partially ordered sets. ACM Transactions on Computational Logic, 17(2):1-35, 2015. Google Scholar
  5. Stefan Bunte and Natalia Kliewer. An overview on vehicle scheduling models. Public Transport, 1(4):299-317, 2009. Google Scholar
  6. Manuel Cáceres. Minimum chain cover in almost linear time. In Proceeding of the 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023), volume 261, pages 31:1-31:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. Google Scholar
  7. Manuel Caceres, Massimo Cairo, Brendan Mumey, Romeo Rizzi, and Alexandru I Tomescu. Minimum path cover in parameterized linear time. arXiv preprint arXiv:2211.09659, 2022. Google Scholar
  8. Manuel Cáceres, Massimo Cairo, Brendan Mumey, Romeo Rizzi, and Alexandru I Tomescu. Sparsifying, shrinking and splicing for minimum path cover in parameterized linear time. In Proceedings of the 33rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2022), pages 359-376. SIAM, 2022. Google Scholar
  9. Ghanshyam Chandra and Chirag Jain. Sequence to graph alignment using gap-sensitive co-linear chaining. In Proceedings of the 27th Annual International Conference on Research in Computational Molecular Biology (RECOMB 2023), pages 58-73. Springer, 2023. Google Scholar
  10. Zheng Chang, Guojun Li, Juntao Liu, Yu Zhang, Cody Ashby, Deli Liu, Carole L Cramer, and Xiuzhen Huang. Bridger: a new framework for de novo transcriptome assembly using RNA-seq data. Genome Biology, 16(1):1-10, 2015. Google Scholar
  11. Li Chen, Rasmus Kyng, Yang P Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva. Maximum flow and minimum-cost flow in almost-linear time. In Proceedings of the 63rd IEEE Annual Symposium on Foundations of Computer Science (FOCS 2022), pages 612-623. IEEE, 2022. Google Scholar
  12. Vasek Chvatal. A greedy heuristic for the set-covering problem. Mathematics of operations research, 4(3):233-235, 1979. Google Scholar
  13. Eleonor Ciurea and Laura Ciupala. Sequential and parallel algorithms for minimum flows. Journal of Applied Mathematics and Computing, 15(1-2):53-75, 2004. Google Scholar
  14. Charles J Colbourn and William R Pulleyblank. Minimizing setups in ordered sets of fixed width. Order, 1(3):225-229, 1985. Google Scholar
  15. Nicola Cotumaccio and Nicola Prezza. On indexing and compressing finite automata. In Proceedings of the 32nd ACM-SIAM Symposium on Discrete Algorithms (SODA 2021), pages 2585-2599. SIAM, 2021. Google Scholar
  16. George Dantzig. Linear programming and extensions. Princeton university press, 1963. Google Scholar
  17. Jacques Desrosiers, Yvan Dumas, Marius M Solomon, and François Soumis. Time constrained routing and scheduling. Handbooks in Operations Research and Management Science, 8:35-139, 1995. Google Scholar
  18. Balázs Dezső, Alpár Jüttner, and Péter Kovács. LEMON-an open source C++ graph template library. Electronic Notes in Theoretical Computer Science, 264(5):23-45, 2011. Google Scholar
  19. Robert P Dilworth. A decomposition theorem for partially ordered sets. Classic Papers in Combinatorics, pages 139-144, 1987. Google Scholar
  20. Yefim Dinitz. Dinitz’ algorithm: The original version and even’s version. In Theoretical Computer Science: Essays in Memory of Shimon Even, pages 218-240. Springer, 2006. Google Scholar
  21. Jack Edmonds and Richard M Karp. Theoretical improvements in algorithmic efficiency for network flow problems. Journal of the ACM, 19(2):248-264, 1972. Google Scholar
  22. Nicholas Eriksson, Lior Pachter, Yumi Mitsuya, Soo-Yon Rhee, Chunlin Wang, Baback Gharizadeh, Mostafa Ronaghi, Robert W Shafer, and Niko Beerenwinkel. Viral population estimation using pyrosequencing. PLoS Computational Biology, 4(5):e1000074, 2008. Google Scholar
  23. Stefan Felsner, Vijay Raghavan, and Jeremy Spinrad. Recognition algorithms for orders of small width and graphs of small Dilworth number. Order, 20(4):351-364, 2003. Google Scholar
  24. Lester Randolph Ford and Delbert R Fulkerson. Maximal flow through a network. Canadian Journal of Mathematics, 8:399-404, 1956. Google Scholar
  25. Delbert R Fulkerson. Note on Dilworth’s decomposition theorem for partially ordered sets. Proceedings of the American Mathematical Society, 7(4):701-702, 1956. Google Scholar
  26. Jakub Gajarskỳ, Petr Hlinenỳ, Daniel Lokshtanov, Jan Obdralek, Sebastian Ordyniak, MS Ramanujan, and Saket Saurabh. FO model checking on posets of bounded width. In Proceedings of the 56th IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS 2015), pages 963-974. IEEE, 2015. Google Scholar
  27. Andrew V Goldberg and Robert E Tarjan. A new approach to the maximum-flow problem. Journal of the ACM, 35(4):921-940, 1988. Google Scholar
  28. Selma Ikiz and Vijay K Garg. Efficient incremental optimal chain partition of distributed program traces. In Proceedings of the 26th IEEE International Conference on Distributed Computing Systems (ICDCS 2006), pages 18-18. IEEE, 2006. Google Scholar
  29. H. V. Jagadish. A compression technique to materialize transitive closure. ACM Transactions on Database Systems, 15(4):558-598, 1990. Google Scholar
  30. Wojciech Jaśkowski and Krzysztof Krawiec. Formal analysis, hardness, and algorithms for extracting internal structure of test-based problems. Evolutionary Computation, 19(4):639-671, 2011. Google Scholar
  31. Arthur B Kahn. Topological sorting of large networks. Communications of the ACM, 5(11):558-562, 1962. Google Scholar
  32. Shimon Kogan and Merav Parter. Beating matrix multiplication for n^1/3-directed shortcuts. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2022. Full version available at URL: https://www.weizmann.ac.il/math/parter/sites/math.parter/files/uploads/main-lipics-full-version_3.pdf.
  33. Shimon Kogan and Merav Parter. Faster and unified algorithms for diameter reducing shortcuts and minimum chain covers. In Proceedings of the 34th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2023), pages 212-239. SIAM, 2023. Google Scholar
  34. Mirosław Kowaluk, Andrzej Lingas, and Johannes Nowak. A path cover technique for LCAs in dags. In Scandinavian Workshop on Algorithm Theory, pages 222-233. Springer, 2008. Google Scholar
  35. Giorgos Kritikakis and Ioannis G Tollis. Fast reachability using DAG decomposition. In Proceedings of the 21st International Symposium on Experimental Algorithms (SEA 2023). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2023. Google Scholar
  36. Panagiotis Lionakis, Giacomo Ortali, and Ioannis G Tollis. Constant-time reachability in DAGs using multidimensional dominance drawings. SN Computer Science, 2(4):320, 2021. Google Scholar
  37. Jun Ma, Manuel Cáceres, Leena Salmela, Veli Mäkinen, and Alexandru I Tomescu. Chaining for Accurate Alignment of Erroneous Long Reads to Acyclic Variation Graphs. Bioinformatics, page btad460, July 2023. URL: https://doi.org/10.1093/bioinformatics/btad460.
  38. Stephen J MacKinnon, Peter D Taylor, Henk Meijer, and Selim G. Akl. An optimal algorithm for assigning cryptographic keys to control access in a hierarchy. IEEE Transactions on Computers, 34(09):797-802, 1985. Google Scholar
  39. Veli Mäkinen, Alexandru I Tomescu, Anna Kuosmanen, Topi Paavilainen, Travis Gagie, and Rayan Chikhi. Sparse dynamic programming on DAGs with small width. ACM Transactions on Algorithms, 15(2):1-21, 2019. Google Scholar
  40. Simeon C Ntafos and S Louis Hakimi. On path cover problems in digraphs and applications to program testing. IEEE Transactions on Software Engineering, 5(5):520-529, 1979. Google Scholar
  41. James B Orlin. A polynomial time primal network simplex algorithm for minimum cost flows. Mathematical Programming, 78:109-129, 1997. Google Scholar
  42. Topi Paavilainen. Faster algorithms for minimum path cover by graph decomposition. Master’s thesis, Helsingin yliopisto, 2018. Google Scholar
  43. Jyotshna Rajput, Ghanshyam Chandra, and Chirag Jain. Co-linear chaining on pangenome graphs. In Proceedings of the 23rd International Workshop on Algorithms in Bioinformatics (WABI 2023), volume 273, pages 12:1-12:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. Google Scholar
  44. Klaus Simon. An improved algorithm for transitive closure on acyclic digraphs. Theoretical Computer Science, 58(1-3):325-346, 1988. Google Scholar
  45. Robert E Tarjan. Edge-disjoint spanning trees and depth-first search. Acta Informatica, 6(2):171-185, 1976. Google Scholar
  46. Alexander I Tomlinson and Vijay K Garg. Monitoring functions on global states of distributed programs. Journal of Parallel and Distributed Computing, 41(2):173-189, 1997. Google Scholar
  47. Cole Trapnell, Brian A Williams, Geo Pertea, Ali Mortazavi, Gordon Kwan, Marijke J Van Baren, Steven L Salzberg, Barbara J Wold, and Lior Pachter. Transcript assembly and quantification by RNA-Seq reveals unannotated transcripts and isoform switching during cell differentiation. Nature Biotechnology, 28(5):511, 2010. Google Scholar
  48. Jan Van Den Brand, Li Chen, Richard Peng, Rasmus Kyng, Yang P Liu, Maximilian Probst Gutenberg, Sushant Sachdeva, and Aaron Sidford. A deterministic almost-linear time algorithm for minimum-cost flow. In Proceedings of the 64th IEEE Annual Symposium on Foundations of Computer Science (FOCS 2023), pages 503-514. IEEE, 2023. Google Scholar
  49. Xianyuan Zhan, Xinwu Qian, and Satish V Ukkusuri. A graph-based approach to measuring the efficiency of an urban taxi service system. IEEE Transactions on Intelligent Transportation Systems, 17(9):2479-2489, 2016. Google Scholar
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