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Fault Tolerant Subgraphs with Applications in Kernelization

Authors William Lochet, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, Roohani Sharma, Meirav Zehavi

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Author Details

William Lochet
  • University of Bergen, Norway
Daniel Lokshtanov
  • University of California, Santa Barbara, USA
Pranabendu Misra
  • Max Planck Institute for Informatics, Saarbrücken, Germany
Saket Saurabh
  • Institute of Mathematical Sciences, HBNI and IRL 2000 ReLaX, Chennai, India
  • University of Bergen, Bergen, Norway
Roohani Sharma
  • Institute of Mathematical Sciences, HBNI, India
Meirav Zehavi
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel

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William Lochet, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, Roohani Sharma, and Meirav Zehavi. Fault Tolerant Subgraphs with Applications in Kernelization. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 47:1-47:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


In the past decade, the design of fault tolerant data structures for networks has become a central topic of research. Particular attention has been given to the construction of a subgraph H of a given digraph D with as fewest arcs/vertices as possible such that, after the failure of any set F of at most k ≥ 1 arcs, testing whether D-F has a certain property P is equivalent to testing whether H-F has that property. Here, reachability (or, more generally, distance preservation) is the most basic requirement to maintain to ensure that the network functions properly. Given a vertex s ∈ V(D), Baswana et al. [STOC'16] presented a construction of H with O(2^kn) arcs in time O(2^{k}nm) where n=|V(D)| and m= |E(D)| such that for any vertex v ∈ V(D): if there exists a path from s to v in D-F, then there also exists a path from s to v in H-F. Additionally, they gave a tight matching lower bound. While the question of the improvement of the dependency on k arises for special classes of digraphs, an arguably more basic research direction concerns the dependency on n (for reachability between a pair of vertices s,t ∈ V(D)) - which are the largest classes of digraphs where the dependency on n can be made sublinear, logarithmic or even constant? Already for the simple classes of directed paths and tournaments, Ω(n) arcs are mandatory. Nevertheless, we prove that "almost acyclicity" suffices to eliminate the dependency on n entirely for a broad class of dense digraphs called bounded independence digraphs. Also, the dependence in k is only a polynomial factor for this class of digraphs. In fact, our sparsification procedure extends to preserve parity-based reachability. Additionally, it finds notable applications in Kernelization: we prove that the classic Directed Feedback Arc Set (DFAS) problem as well as Directed Edge Odd Cycle Transversal (DEOCT) (which, in sharp contrast to DFAS, is W[1]-hard on general digraphs) admit polynomial kernels on bounded independence digraphs. In fact, for any p ∈ N, we can design a polynomial kernel for the problem of hitting all cycles of length ℓ where (ℓ mod p = 1). As a complementary result, we prove that DEOCT is NP-hard on tournaments by establishing a combinatorial identity between the minimum size of a feedback arc set and the minimum size of an edge odd cycle transversal. In passing, we also improve upon the running time of the sub-exponential FPT algorithm for DFAS in digraphs of bounded independence number given by Misra et at. [FSTTCS 2018], and give the first sub-exponential FPT algorithm for DEOCT in digraphs of bounded independence number.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Sparsification and spanners
  • sparsification
  • kernelization
  • fault tolerant subgraphs
  • directed feedback arc set
  • directed edge odd cycle transversal
  • bounded independence number digraphs


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  1. Open Problems in Parameterized Complexity. Accessed: 2019-02-15.
  2. Nir Ailon, Moses Charikar, and Alantha Newman. Aggregating inconsistent information: Ranking and clustering. J. ACM, 55(5):23:1-23:27, 2008. Google Scholar
  3. Noga Alon. Ranking Tournaments. SIAM J. Discrete Math., 20(1):137-142, 2006. Google Scholar
  4. Noga Alon, Daniel Lokshtanov, and Saket Saurabh. Fast FAST. In Automata, Languages and Programming, 36th International Colloquium, ICALP 2009, Rhodes, Greece, July 5-12, 2009, Proceedings, Part I, pages 49-58, 2009. Google Scholar
  5. Alexandr Andoni, Anupam Gupta, and Robert Krauthgamer. Towards (1+ ε)-approximate flow sparsifiers. In Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms, pages 279-293. Society for Industrial and Applied Mathematics, 2014. Google Scholar
  6. Jørgen Bang-Jensen and Carsten Thomassen. A Polynomial Algorithm for the 2-Path Problem for Semicomplete Digraphs. SIAM J. Discrete Math., 5(3):366-376, 1992. Google Scholar
  7. Surender Baswana, Keerti Choudhary, Moazzam Hussain, and Liam Roditty. Approximate single source fault tolerant shortest path. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1901-1915. SIAM, 2018. Google Scholar
  8. Surender Baswana, Keerti Choudhary, and Liam Roditty. Fault tolerant reachability for directed graphs. In International Symposium on Distributed Computing, pages 528-543. Springer, 2015. Google Scholar
  9. Surender Baswana, Keerti Choudhary, and Liam Roditty. Fault-Tolerant Subgraph for Single-Source Reachability: General and Optimal. SIAM Journal on Computing, 47(1):80-95, 2018. Google Scholar
  10. Stéphane Bessy, Fedor V Fomin, Serge Gaspers, Christophe Paul, Anthony Perez, Saket Saurabh, and Stéphan Thomassé. Kernels for feedback arc set in tournaments. Journal of Computer and System Sciences, 77(6):1071-1078, 2011. Google Scholar
  11. Davide Bilò, Fabrizio Grandoni, Luciano Gualà, Stefano Leucci, and Guido Proietti. Improved purely additive fault-tolerant spanners. In Algorithms-ESA 2015, pages 167-178. Springer, 2015. Google Scholar
  12. Davide Bilò, Luciano Gualà, Stefano Leucci, and Guido Proietti. Fault-tolerant approximate shortest-path trees. In European Symposium on Algorithms, pages 137-148. Springer, 2014. Google Scholar
  13. Diptarka Chakraborty and Debarati Das. Sparse Weight Tolerant Subgraph for Single Source Shortest Path. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  14. Pierre Charbit, Stéphan Thomassé, and Anders Yeo. The Minimum Feedback Arc Set Problem is NP-Hard for Tournaments. Combinatorics, Probability & Computing, 16(1):1-4, 2007. Google Scholar
  15. Moses Charikar, Tom Leighton, Shi Li, and Ankur Moitra. Vertex sparsifiers and abstract rounding algorithms. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, pages 265-274. IEEE, 2010. Google Scholar
  16. Shiri Chechik. Fault-tolerant compact routing schemes for general graphs. In International Colloquium on Automata, Languages, and Programming, pages 101-112. Springer, 2011. Google Scholar
  17. Shiri Chechik, Michael Langberg, David Peleg, and Liam Roditty. Fault tolerant spanners for general graphs. SIAM Journal on Computing, 39(7):3403-3423, 2010. Google Scholar
  18. Shiri Chechik, Michael Langberg, David Peleg, and Liam Roditty. F-sensitivity distance oracles and routing schemes. Algorithmica, 63(4):861-882, 2012. Google Scholar
  19. Jianer Chen, Yang Liu, Songjian Lu, Barry O’sullivan, and Igor Razgon. A fixed-parameter algorithm for the directed feedback vertex set problem. Journal of the ACM (JACM), 55(5):21, 2008. Google Scholar
  20. Rajesh Chitnis. Directed Graphs: Fixed-Parameter Tractability & Beyond. PhD thesis, University of Maryland, 2014. Google Scholar
  21. Rajesh Chitnis and Mohammad Taghi Hajiaghayi. Shadowless solutions for fixed-parameter tractability of directed graphs. Encyclopedia of Algorithms, pages 1-5, 2008. Google Scholar
  22. Julia Chuzhoy. On vertex sparsifiers with Steiner nodes. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 673-688. ACM, 2012. Google Scholar
  23. Marek Cygan, Fedor V Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized algorithms, volume 4. Springer, 2015. Google Scholar
  24. Erik Demaine, Gregory Z Gutin, Dániel Marx, and Ulrike Stege. 07281 Open Problems-Structure Theory and FPT Algorithmcs for Graphs, Digraphs and Hypergraphs. In Dagstuhl Seminar Proceedings. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2007. Google Scholar
  25. Reinhard Diestel. Graph Theory, volume 173 of. Graduate texts in mathematics, page 7, 2012. Google Scholar
  26. Michael Dinitz and Robert Krauthgamer. Fault-tolerant spanners: better and simpler. In Proceedings of the 30th annual ACM SIGACT-SIGOPS symposium on Principles of distributed computing, pages 169-178. ACM, 2011. Google Scholar
  27. Michael Dom, Jiong Guo, Falk Hüffner, Rolf Niedermeier, and Anke Truß. Fixed-parameter tractability results for feedback set problems in tournaments. In Italian Conference on Algorithms and Complexity, pages 320-331. Springer, 2006. Google Scholar
  28. Rodney G Downey and Michael R Fellows. Fundamentals of parameterized complexity, volume 4. Springer, 2013. Google Scholar
  29. Matthias Englert, Anupam Gupta, Robert Krauthgamer, Harald Racke, Inbal Talgam-Cohen, and Kunal Talwar. Vertex sparsifiers: New results from old techniques. SIAM Journal on Computing, 43(4):1239-1262, 2014. Google Scholar
  30. Uriel Feige. Faster FAST(Feedback Arc Set in Tournaments). CoRR, abs/0911.5094, 2009. URL:
  31. Paola Festa, Panos M Pardalos, and Mauricio GC Resende. Feedback set problems. In Handbook of combinatorial optimization, pages 209-258. Springer, 1999. Google Scholar
  32. Fedor V Fomin and Michał Pilipczuk. Subexponential parameterized algorithm for computing the cutwidth of a semi-complete digraph. In European Symposium on Algorithms, pages 505-516. Springer, 2013. Google Scholar
  33. Alexandra Fradkin and Paul Seymour. Edge-disjoint paths in digraphs with bounded independence number. Journal of Combinatorial Theory, Series B, 110:19-46, 2015. Google Scholar
  34. Georges Gardarin and Wesley W. Chu. Integrity of databases: A general lockout algorithm with deadlock avoidance. In Proceedings of the IFIP Working Conference on Modelling in Data Base Management Systems, [Freudenstadt, Germany, 5-8 January 1976], pages 395-411. North-Holland Publishing Company, 1976. Google Scholar
  35. Richard M Karp. Reducibility among combinatorial problems. In Complexity of computer computations, pages 85-103. Springer, 1972. Google Scholar
  36. Marek Karpinski and Warren Schudy. Faster Algorithms for Feedback Arc Set Tournament, Kemeny Rank Aggregation and Betweenness Tournament. In Algorithms and Computation - 21st International Symposium, ISAAC 2010, Jeju Island, Korea, December 15-17, 2010, Proceedings, Part I, pages 3-14, 2010. Google Scholar
  37. Claire Kenyon-Mathieu and Warren Schudy. How to rank with few errors. In STOC, volume 7, pages 95-103, 2007. Google Scholar
  38. F Thomson Leighton and Ankur Moitra. Extensions and limits to vertex sparsification. In Proceedings of the forty-second ACM symposium on Theory of computing, pages 47-56. ACM, 2010. Google Scholar
  39. Daniel Lokshtanov, MS Ramanujan, and Saket Saurabh. When recursion is better than iteration: A linear-time algorithm for acyclicity with few error vertices. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1916-1933. SIAM, 2018. Google Scholar
  40. Daniel Lokshtanov, MS Ramanujan, Saket Saurabh, and Meirav Zehavi. Parameterized complexity and approximability of directed odd cycle transversal. arXiv preprint, 2017. URL:
  41. Dániel Marx. Some open problems in parameterized complexity y. dmarx/papers/marx-dagstuhl2017-open.pdf, 2-17.
  42. Dániel Marx. What’s next? Future directions in parameterized complexity. In The Multivariate Algorithmic Revolution and Beyond, pages 469-496. Springer, 2012. Google Scholar
  43. Pranabendu Misra, Saket Saurabh, Roohani Sharma, and Meirav Zehavi. Sub-Exponential Time Parameterized Algorithms for Graph Layout Problems on Digraphs with Bounded Independence Number. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  44. Merav Parter. Dual failure resilient BFS structure. In Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, pages 481-490. ACM, 2015. Google Scholar
  45. Merav Parter. Vertex fault tolerant additive spanners. Distributed Computing, 30(5):357-372, 2017. Google Scholar
  46. Merav Parter et al. Fault-tolerant logical network structures. Bulletin of EATCS, 1(118), 2016. Google Scholar
  47. Merav Parter and David Peleg. Sparse fault-tolerant BFS trees. In European Symposium on Algorithms, pages 779-790. Springer, 2013. Google Scholar
  48. Abraham Silberschatz, Greg Gagne, and Peter B Galvin. Operating system concepts (9th edition). Wiley, 2012. Google Scholar
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