Efficient algorithms for measuring the funnel-likeness of DAGs

  • 38 Accesses


We propose funnels as a new natural subclass of DAGs. Intuitively, a DAG is a funnel if every source-sink path can be uniquely identified by one of its arcs. Funnels are an analogue to trees for directed graphs, being more restrictive than DAGs but more expressive than mere in-/out-trees. Computational problems such as finding vertex-disjoint paths or tracking the origin of memes remain NP-hard on DAGs while on funnels they become solvable in polynomial time. Our main focus is the algorithmic complexity of finding out how funnel-like a given DAG is. To this end, we identify the NP-hard problem of computing the arc-deletion distance of a given DAG to a funnel. We develop efficient exact and approximation algorithms for the problem and test them on synthetic random graphs and real-world graphs.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 99

This is the net price. Taxes to be calculated in checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9


  1. 1.

    Clearly, the problem can also be defined for general digraphs as input. In this work, however, we mainly focus on the case where the input is a DAG. Whenever results transfer to the general case we state them explicitly.

  2. 2.

    Informally, the Exponential Time Hypothesis states that there is no algorithm for 3SAT  that runs in \(2^{o(n+m)}\cdot (n+m)^{O(1)}\) time, where n is the number of variables and m is the number of clauses in the formula (Impagliazzo and Paturi 2001; Impagliazzo et al. 2001).

  3. 3.

    To avoid confusion with the vertices of D, we refer to vertices of T as nodes.

  4. 4.

    Listed at and obtained using pacman.


  1. Agrawal A, Saurabh S, Sharma R, Zehavi M (2018) Kernels for deletion to classes of acyclic digraphs. J Comput Syst Sci 92:9–21

  2. Agrawal A, Saurabh S, Sharma R, Zehavi M (2018) Parameterised algorithms for deletion to classes of DAGs. Theory Comput Syst 62(8):1880–1909

  3. Ailon N, Alon N (2007) Hardness of fully dense problems. Inf Comput 205(8):1117–1129

  4. Bang-Jensen J, Gutin G (2018) Classes of directed graphs. Springer Monographs in Mathematics Springer. Springer, Berlin

  5. Bang-Jensen J, Gutin GZ (2009) Digraphs: theory, algorithms and applications. Springer, Berlin

  6. Bessy S, Fomin FV, Gaspers S, Paul C, Perez A, Saurabh S, Thomassé S (2011) Kernels for feedback arc set in tournaments. J Comput Syst Sci 77(6):1071–1078

  7. Bonsma P, Lokshtanov D (2011) Feedback vertex set in mixed graphs. In: Proceedings of the 12th international symposium on algorithms and data structures (WADS ’11), LNCS, vol 6844, pp 122–133. Springer, Berlin

  8. Cai L (2003) Parameterized complexity of vertex colouring. Discrete Appl Math 127(3):415–429

  9. Charbit P, Thomassé S, Yeo A (2007) The minimum feedback arc set problem is NP-hard for tournaments. Combin Probab Comput 16(1):1–4

  10. Chen J, Liu Y, Lu S, O’Sullivan B, Razgon I (2008) A fixed-parameter algorithm for the directed feedback vertex set problem. J ACM 55(5):21

  11. Chitnis R, Cygan M, Hajiaghayi M, Marx D (2015) Directed subset feedback vertex set is fixed-parameter tractable. ACM Trans Algorithms 11(4):28

  12. Cygan M, Fomin FV, Łukasz K, Marx DLD, Pilipczuk M, Pilipczuk M, Saurabh S (2015) Parametr Algorithms. Springer International Publishing, Berlin

  13. Diestel R (2016) Graph theory. Graduate texts in mathematics, vol 173, 5th edn. Springer, Belrin

  14. Dom M, Guo J, Hüffner F, Niedermeier R, Truß A (2010) Fixed-parameter tractability results for feedback set problems in tournaments. J Discrete Algorithms 8(1):76–86

  15. Downey RG, Fellows MR (2013) Fundamentals of parameterized complexity. Texts in computer science. Springer, Berlin

  16. Feige U (1998) A threshold of \(\ln n\) for approximating set cover. J ACM 45(4):634–652

  17. Fortune S, Hopcroft J, Wyllie J (1980) The directed subgraph homeomorphism problem. Theor Comput Sci 10(2):111–121

  18. Ganian R, Hlinený P, Kneis J, Langer A, Obdrzálek J, Rossmanith P (2014) Digraph width measures in parameterized algorithmics. Discrete Appl Math 168:88–107

  19. Ganian R, Hlinený P, Kneis J, Meister D, Obdrzálek J, Rossmanith P, Sikdar S (2016) Are there any good digraph width measures? J Combin Theory Ser B 116:250–286

  20. Guo J, Hüffner F, Niedermeier R (2004) A structural view on parameterizing problems: distance from triviality. In: Proceedings of the 1st international workshop on parameterized and exact computation (IWPEC ’04), pp 162–173. Springer, Berlin

  21. Impagliazzo R, Paturi R (2001) On the complexity of \(k\)-SAT. J Comput Syst Sci 62(2):367–375

  22. Impagliazzo R, Paturi R, Zane F (2001) Which problems have strongly exponential complexity? J Comput Syst Sci 63(4):512–530

  23. Johnson T, Robertson N, Seymour PD, Thomas R (2001) Directed tree-width. J Combin Theory Ser B 82(1):138–154

  24. Kenyon-Mathieu C, Schudy W (2007) How to rank with few errors. In: Proceedings of the 39th ACM symposium on theory of computing (STOC ’07), pp 95–103. ACM

  25. Kunegis J (2013) KONECT—The Koblenz network collection. In: Proceedings of the 22nd international world wide web conference (WWW ’13), pp 1343–1350. ACM

  26. Lehmann J (2017) The computational complexity of worst case flows in unreliable flow networks. Bachelor thesis, Institut für Theoretische Informatik, Universität zu Lübeck

  27. Leskovec J, Backstrom L, Kleinberg J (2009) Meme-tracking and the dynamics of the news cycle. In: Proceedings of the 15th ACM SIGKDD international conference on knowledge discovery and data mining (KDD ’09), pp 497–506. ACM

  28. Lund C, Yannakakis M (1993) The approximation of maximum subgraph problems. In: Proceedings of the 20th international colloquium on automata, languages, and programming (ICALP ’93), LNCS, vol 700, pp 40–51. Springer

  29. Millani MG (2017) Funnels—algorithmic complexity of problems on special directed acyclic graphs. Master thesis, Algorithmics and computational complexity (AKT), TU Berlin.

  30. Millani MG (2017) Parfunn—parameters for funnels.

  31. Millani MG, Molter H, Niedermeier R, Sorge M (2018) Efficient algorithms for measuring the funnel-likeness of DAGs. In: Proceedings of the 5th international symposium on combinatorial optimization (ISCO ’18), LNCS, vol 10856, pp 183–195. Springer

  32. Mnich M, van Leeuwen EJ (2017) Polynomial kernels for deletion to classes of acyclic digraphs. Discrete Optim 25:48–76

  33. Niedermeier R (2010) Reflections on multivariate algorithmics and problem parameterization. In: Proceedings of the 27th international symposium on theoretical aspects of computer science (STACS ’10), pp 17–32. Schloss Dagstuhl–Leibniz-Zentrum für Informatik

  34. van Bevern R, Bredereck R, Chopin M, Hartung S, Hüffner F, Nichterlein A, Suchý O (2017) Fixed-parameter algorithms for DAG partitioning. Discrete Appl Math 220:134–160

Download references


We are grateful to anonymous reviewers of Journal of Combinatorial Optimization whose constructive feedback helped to improve the presentation and remove some bugs from this paper.

Author information

Correspondence to Hendrik Molter.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

MGM was partially supported by the DFG, Project FPTinP (NI 369/16). HM was partially supported by the DFG, Project MATE (NI 369/17). MS was partially supported by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA Grant Agreement Number 631163.11, by the Israel Science Foundation (Grant No. 551145/14), and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under Grant Agreement Number 714704.

An extended abstract of this work appeared in the proceedings of the 5th International Symposium on Combinatorial Optimization (ISCO ’18) (Millani et al. 2018). This version contains all proof details, additional inapproximability results, and extended experimental findings. Work started while all authors were with TU Berlin. The main work of MS was carried out while with Ben-Gurion University of the Negev, Beer Sheva, Israel.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Garlet Millani, M., Molter, H., Niedermeier, R. et al. Efficient algorithms for measuring the funnel-likeness of DAGs. J Comb Optim 39, 216–245 (2020) doi:10.1007/s10878-019-00464-4

Download citation


  • Directed graphs
  • Acyclic digraph
  • NP-hard problems
  • Approximation hardness
  • Fixed-parameter tractability
  • Approximation algorithms
  • Graph parameters
  • Experiments