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Width Helps and Hinders Splitting Flows

Authors Manuel Cáceres , Massimo Cairo, Andreas Grigorjew , Shahbaz Khan , Brendan Mumey , Romeo Rizzi, Alexandru I. Tomescu , Lucia Williams



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Author Details

Manuel Cáceres
  • Department of Computer Science, University of Helsinki, Finland
Massimo Cairo
  • Department of Computer Science, University of Helsinki, Finland
Andreas Grigorjew
  • Department of Computer Science, University of Helsinki, Finland
Shahbaz Khan
  • Department of Computer Science and Engineering, Indian Institute of Technology Roorkee, India
Brendan Mumey
  • School of Computing, Montana State University, Bozeman, MT, USA
Romeo Rizzi
  • Department of Computer Science, University of Verona, Italy
Alexandru I. Tomescu
  • Department of Computer Science, University of Helsinki, Finland
Lucia Williams
  • School of Computing, Montana State University, Bozeman, MT, USA

Cite AsGet BibTex

Manuel Cáceres, Massimo Cairo, Andreas Grigorjew, Shahbaz Khan, Brendan Mumey, Romeo Rizzi, Alexandru I. Tomescu, and Lucia Williams. Width Helps and Hinders Splitting Flows. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 31:1-31:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ESA.2022.31

Abstract

Minimum flow decomposition (MFD) is the NP-hard problem of finding a smallest decomposition of a network flow X on directed graph G into weighted source-to-sink paths whose superposition equals X. We focus on a common formulation of the problem where the path weights must be non-negative integers and also on a new variant where these weights can be negative. We show that, for acyclic graphs, considering the width of the graph (the minimum number of s-t paths needed to cover all of its edges) yields advances in our understanding of its approximability. For the non-negative version, we show that a popular heuristic is a O(log |X|)-approximation (|X| being the total flow of X) on graphs satisfying two properties related to the width (satisfied by e.g., series-parallel graphs), and strengthen its worst-case approximation ratio from Ω(√m) to Ω(m / log m) for sparse graphs, where m is the number of edges in the graph. For the negative version, we give a (⌈log ║X║⌉+1)-approximation (║X║ being the maximum absolute value of X on any edge) using a power-of-two approach, combined with parity fixing arguments and a decomposition of unitary flows (║X║ ≤ 1) into at most width paths. We also disprove a conjecture about the linear independence of minimum (non-negative) flow decompositions posed by Kloster et al. [ALENEX 2018], but show that its useful implication (polynomial-time assignments of weights to a given set of paths to decompose a flow) holds for the negative version.

Subject Classification

ACM Subject Classification
  • Theory of computation → Network flows
Keywords
  • Flow decomposition
  • approximation algorithms
  • graph width

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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. Jasmijn A Baaijens, Leen Stougie, and Alexander Schönhuth. Strain-aware assembly of genomes from mixed samples using flow variation graphs. In International Conference on Research in Computational Molecular Biology, pages 221-222. Springer, 2020. Google Scholar
  3. Elsa Bernard, Laurent Jacob, Julien Mairal, and Jean-Philippe Vert. Efficient RNA isoform identification and quantification from RNA-Seq data with network flows. Bioinformatics, 30(17):2447-2455, 2014. Google Scholar
  4. Dimitris Bertsimas, Ebrahim Nasrabadi, and Sebastian Stiller. Robust and adaptive network flows. Operations Research, 61(5):1218-1242, 2013. URL: https://doi.org/10.1287/opre.2013.1200.
  5. Manuel Cáceres, Massimo Cairo, Andreas Grigorjew, Shahbaz Khan, Brendan Mumey, Romeo Rizzi, Alexandru I. Tomescu, and Lucia Williams. Width helps and hinders splitting flows. CoRR, 2022. URL: https://doi.org/10.48550/ARXIV.2207.02136.
  6. 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 SODA 2022 - ACM-SIAM Symposium on Discrete Algorithms, pages 359-376, 2022. URL: https://doi.org/10.1137/1.9781611977073.18.
  7. Fernando H. C. Dias, Lucia Williams, Brendan Mumey, and Alexandru I. Tomescu. Fast, Flexible, and Exact Minimum Flow Decompositions via ILP. arXiv, arXiv:2201.10923, 2022. To appear in the Proceedings of RECOMB 2022 - 26th Annual International Conference on Research in Computational Molecular Biology. URL: http://arxiv.org/abs/2201.10923.
  8. Robert P Dilworth. A decomposition theorem for partially ordered sets. Annals of Mathematics, 51(1):161-166, 1950. URL: http://www.jstor.org/stable/1969503.
  9. David Eppstein. Parallel recognition of series-parallel graphs. Information and Computation, 98(1):41-55, 1992. URL: https://doi.org/10.1016/0890-5401(92)90041-D.
  10. 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
  11. Thomas Gatter and Peter F Stadler. Ryūtō: network-flow based transcriptome reconstruction. BMC bioinformatics, 20(1):1-14, 2019. Google Scholar
  12. Tzvika Hartman, Avinatan Hassidim, Haim Kaplan, Danny Raz, and Michal Segalov. How to split a flow? In 2012 Proceedings IEEE INFOCOM, pages 828-836. IEEE, 2012. Google Scholar
  13. A. Jain and N. Chandrasekharan. An efficient parallel algorithm for min-cost flow on directed series-parallel networks. In Proceedings Seventh International Parallel Processing Symposium, pages 188-192, 1993. URL: https://doi.org/10.1109/IPPS.1993.262879.
  14. Kyle Kloster, Philipp Kuinke, Michael P O'Brien, Felix Reidl, Fernando Sánchez Villaamil, Blair D Sullivan, and Andrew van der Poel. A practical fpt algorithm for flow decomposition and transcript assembly. In 2018 Proceedings of the Twentieth Workshop on Algorithm Engineering and Experiments (ALENEX), pages 75-86. SIAM, 2018. Google Scholar
  15. Brendan Mumey, Samareh Shahmohammadi, Kathryn McManus, and Sean Yaw. Parity balancing path flow decomposition and routing. In 2015 IEEE Globecom Workshops (GC Wkshps), pages 1-6. IEEE, 2015. Google Scholar
  16. Nils Olsen, Natalia Kliewer, and Lena Wolbeck. A study on flow decomposition methods for scheduling of electric buses in public transport based on aggregated time-space network models. Central European Journal of Operations Research, 2020. URL: https://doi.org/10.1007/s10100-020-00705-6.
  17. James B. Orlin. Max flows in O(nm) time, or better. In Dan Boneh, Tim Roughgarden, and Joan Feigenbaum, editors, Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 765-774. ACM, 2013. URL: https://doi.org/10.1145/2488608.2488705.
  18. Mihaela Pertea, Geo M Pertea, Corina M Antonescu, Tsung-Cheng Chang, Joshua T Mendell, and Steven L Salzberg. StringTie enables improved reconstruction of a transcriptome from RNA-seq reads. Nature biotechnology, 33(3):290-295, 2015. Google Scholar
  19. Alexander Schrijver. Theory of Linear and Integer Programming. John Wiley & Sons, Inc., 1986. Google Scholar
  20. Mingfu Shao and Carl Kingsford. Theory and a heuristic for the minimum path flow decomposition problem. IEEE/ACM transactions on computational biology and bioinformatics, 16(2):658-670, 2017. Google Scholar
  21. Mingxiang Teng, Michael I Love, Carrie A Davis, Sarah Djebali, Alexander Dobin, Brenton R Graveley, Sheng Li, Christopher E Mason, Sara Olson, Dmitri Pervouchine, et al. A benchmark for rna-seq quantification pipelines. Genome biology, 17(1):1-12, 2016. Google Scholar
  22. Alexandru I Tomescu, Travis Gagie, Alexandru Popa, Romeo Rizzi, Anna Kuosmanen, and Veli Mäkinen. Explaining a weighted DAG with few paths for solving genome-guided multi-assembly. IEEE/ACM transactions on computational biology and bioinformatics, 12(6):1345-1354, 2015. Google Scholar
  23. Alexandru I Tomescu, Anna Kuosmanen, Romeo Rizzi, and Veli Mäkinen. A novel min-cost flow method for estimating transcript expression with RNA-Seq. In BMC bioinformatics, volume 14, pages S15:1-S15:10. Springer, 2013. Google Scholar
  24. Jacobo Valdes, Robert E. Tarjan, and Eugene L. Lawler. The recognition of series parallel digraphs. SIAM Journal on Computing, 11(2):298-313, 1982. URL: https://doi.org/10.1137/0211023.
  25. Benedicte Vatinlen, Fabrice Chauvet, Philippe Chrétienne, and Philippe Mahey. Simple bounds and greedy algorithms for decomposing a flow into a minimal set of paths. European Journal of Operational Research, 185(3):1390-1401, 2008. Google Scholar
  26. Lucia Williams, Gillian Reynolds, and Brendan Mumey. RNA Transcript Assembly Using Inexact Flows. In 2019 IEEE International Conference on Bioinformatics and Biomedicine (BIBM), pages 1907-1914. IEEE, 2019. Google Scholar
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