Fixed-Parameter Sensitivity Oracles

Authors Davide Bilò , Katrin Casel , Keerti Choudhary , Sarel Cohen , Tobias Friedrich , J.A. Gregor Lagodzinski , Martin Schirneck, Simon Wietheger

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Davide Bilò
  • Department of Humanities and Social Sciences, University of Sassari, Italy
Katrin Casel
  • Hasso Plattner Institute, University of Potsdam, Germany
Keerti Choudhary
  • Department of Computer Science and Engineering, Indian Institute of Technology Delhi, India
Sarel Cohen
  • School of Computer Science, Tel-Aviv-Yaffo Academic College, Israel
Tobias Friedrich
  • Hasso Plattner Institute, University of Potsdam, Germany
J.A. Gregor Lagodzinski
  • Hasso Plattner Institute, University of Potsdam, Germany
Martin Schirneck
  • Hasso Plattner Institute, University of Potsdam, Germany
Simon Wietheger
  • Hasso Plattner Institute, University of Potsdam, Germany

Cite AsGet BibTex

Davide Bilò, Katrin Casel, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, J.A. Gregor Lagodzinski, Martin Schirneck, and Simon Wietheger. Fixed-Parameter Sensitivity Oracles. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We combine ideas from distance sensitivity oracles (DSOs) and fixed-parameter tractability (FPT) to design sensitivity oracles for FPT graph problems. An oracle with sensitivity f for an FPT problem Π on a graph G with parameter k preprocesses G in time O(g(f,k) ⋅ poly(n)). When queried with a set F of at most f edges of G, the oracle reports the answer to the Π - with the same parameter k - on the graph G-F, i.e., G deprived of F. The oracle should answer queries in a time that is significantly faster than merely running the best-known FPT algorithm on G-F from scratch. We design sensitivity oracles for the k-Path and the k-Vertex Cover problem. Our first oracle for k-Path has size O(k^{f+1}) and query time O(f min{f, log(f) + k}). We use a technique inspired by the work of Weimann and Yuster [FOCS 2010, TALG 2013] on distance sensitivity problems to reduce the space to O(({f+k}/f)^f ({f+k}/k)^k fk⋅log(n)) at the expense of increasing the query time to O(({f+k}/f)^f ({f+k}/k)^k f min{f,k}⋅log(n)). Both oracles can be modified to handle vertex-failures, but we need to replace k with 2k in all the claimed bounds. Regarding k-Vertex Cover, we design three oracles offering different trade-offs between the size and the query time. The first oracle takes O(3^{f+k}) space and has O(2^f) query time, the second one has a size of O(2^{f+k²+k}) and a query time of O(f+k²); finally, the third one takes O(fk+k²) space and can be queried in time O(1.2738^k + f). All our oracles are computable in time (at most) proportional to their size and the time needed to detect a k-path or k-vertex cover, respectively. We also provide an interesting connection between k-Vertex Cover and the fault-tolerant shortest path problem, by giving a DSO of size O(poly(f,k) ⋅ n) with query time in O(poly(f,k)), where k is the size of a vertex cover. Following our line of research connecting fault-tolerant FPT and shortest paths problems, we introduce parameterization to the computation of distance preservers. We study the problem, given a directed unweighted graph with a fixed source s and parameters f and k, to construct a polynomial-sized oracle that efficiently reports, for any target vertex v and set F of at most f edges, whether the distance from s to v increases at most by an additive term of k in G-F. The oracle size is O(2^k k²⋅n), while the time needed to answer a query is O(2^k f^ω k^ω), where ω < 2.373 is the matrix multiplication exponent. The second problem we study is about the construction of bounded-stretch fault-tolerant preservers. We construct a subgraph with O(2^{fk+f+k} k ⋅ n) edges that preserves those s-v-distances that do not increase by more than k upon failure of F. This improves significantly over the Õ(f n^{2-1/(2^f)}) bound in the unparameterized case by Bodwin et al. [ICALP 2017].

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Graph algorithms
  • data structures
  • distance preservers
  • distance sensitivity oracles
  • fault tolerance
  • fixed-parameter tractability
  • k-path
  • vertex cover


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  1. Abu Reyan Ahmed, Greg Bodwin, Faryad Darabi Sahneh, Keaton Hamm, Mohammad Javad Latifi Jebelli, Stephen G. Kobourov, and Richard Spence. Graph spanners: A tutorial review. Computer Science Review, 37:100253, 2020. URL:
  2. Josh Alman, Matthias Mnich, and Virginia Vassilevska Williams. Dynamic Parameterized Problems and Algorithms. ACM Transactions on Algorithms, 16:45:1-45:46, 2020. URL:
  3. Josh Alman and Virginia Vassilevska Williams. A Refined Laser Method and Faster Matrix Multiplication. In Proceedings of the 32nd Symposium on Discrete Algorithms (SODA), pages 522-539, 2021. URL:
  4. Surender Baswana, Keerti Choudhary, and Liam Roditty. Fault Tolerant Subgraph for Single Source Reachability: Generic and Optimal. In Proceedings of the 48th Symposium on Theory of Computing (STOC), pages 509-518, 2016. URL:
  5. Ivona Bezáková, Radu Curticapean, Holger Dell, and Fedor V. Fomin. Finding Detours is Fixed-Parameter Tractable. SIAM Journal on Discrete Mathematics, 33:2326-2345, 2019. URL:
  6. Davide Bilò, Sarel Cohen, Tobias Friedrich, and Martin Schirneck. Near-Optimal Deterministic Single-Source Distance Sensitivity Oracles. In Proceedings of the 29th European Symposium on Algorithms (ESA), pages 18:1-18:17, 2021. URL:
  7. Davide Bilò, Sarel Cohen, Tobias Friedrich, and Martin Schirneck. Space-Efficient Fault-Tolerant Diameter Oracles. In Proceedings of the 46th International Symposium on Mathematical Foundations of Computer Science (MFCS), pages 18:1-18:16, 2021. URL:
  8. Greg Bodwin, Michael Dinitz, and Caleb Robelle. Optimal Vertex Fault-Tolerant Spanners in Polynomial Time. In Proceedings of the 32nd Symposium on Discrete Algorithms (SODA), pages 2924-2938, 2021. URL:
  9. Greg Bodwin, Fabrizio Grandoni, Merav Parter, and Virginia Vassilevska Williams. Preserving Distances in Very Faulty Graphs. In Proceedings of the 44th International Colloquium on Automata, Languages, and Programming, (ICALP), pages 73:1-73:14, 2017. URL:
  10. Jonathan F. Buss and Judy Goldsmith. Nondeterminism Within P. SIAM Journal on Computing, 22:560-572, 1993. URL:
  11. Shiri Chechik, Sarel Cohen, Amos Fiat, and Haim Kaplan. (1+ε)-Approximate f-Sensitive Distance Oracles. In Proceedings of the 28th Symposium on Discrete Algorithms (SODA), pages 1479-1496, 2017. URL:
  12. Jianer Chen, Iyad A. Kanj, and Ge Xia. Improved Upper Bounds for Vertex Cover. Theoretical Computer Science, 411:3736-3756, 2010. URL:
  13. Jiehua Chen, Wojciech Czerwinski, Yann Disser, Andreas E. Feldmann, Danny Hermelin, Wojciech Nadara, Marcin Pilipczuk, Michał Pilipczuk, Manuel Sorge, Bartlomiej Wróblewski, and Anna Zych-Pawlewicz. Efficient Fully Dynamic Elimination Forests with Applications to Detecting Long Paths and Cycles. In Proceedings of the 32nd Symposium on Discrete Algorithms (SODA), pages 796-809, 2021. URL:
  14. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, Cham, Switzerland, 2015. URL:
  15. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, London, UK, 2013. URL:
  16. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Long Directed (s,t)-path: FPT Algorithm. Information Processing Letters, 140:8-12, 2018. URL:
  17. Fedor V. Fomin, Daniel Lokshtanov, and Saket Saurabh. Efficient Computation of Representative Sets with Applications in Parameterized and Exact Algorithms. In Proceedings of the 25th Symposium on Discrete Algorithms (SODA), pages 142-151, 2014. URL:
  18. Fabrizio Grandoni and Virginia Vassilevska Williams. Faster Replacement Paths and Distance Sensitivity Oracles. ACM Transaction on Algorithms, 16:15:1-15:25, 2020. URL:
  19. Yong Gu and Hanlin Ren. Constructing a Distance Sensitivity Oracle in O(n^2.5794M) Time. In Proceedings of the 48th International Colloquium on Automata, Languages, and Programming (ICALP), pages 76:1-76:20, 2021. URL:
  20. Torben Hagerup, Peter Bro Miltersen, and Rasmus Pagh. Deterministic Dictionaries. Journal of Algorithms, 41:69-85, 2001. URL:
  21. Yoichi Iwata and Keigo Oka. Fast Dynamic Graph Algorithms for Parameterized Problems. In Proceedings of the 14th Scandinavian Workshop on Algorithm Theory (SWAT), pages 241-252, 2014. URL:
  22. William Lochet, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, Roohani Sharma, and Meirav Zehavi. Fault Tolerant Subgraphs with Applications in Kernelization. In Proceeings of the 11th Innovations in Theoretical Computer Science Conference (ITCS), pages 47:1-47:22, 2020. URL:
  23. Pranabendu Misra. On Fault Tolerant Feedback Vertex Set. CoRR, abs/2009.06063, 2020. ArXiv preprint. URL:
  24. Michael Mitzenmacher and Eli Upfal. Probability and Computing. Cambridge University Press, New York, NY, USA, 2nd edition, 2017. Google Scholar
  25. Rolf Niedermeier. Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford, UK, 2006. URL:
  26. Merav Parter. Dual Failure Resilient BFS Structure. In Proceedings of the 2015 Symposium on Principles of Distributed Computing (PODC), pages 481-490, 2015. URL:
  27. Merav Parter and David Peleg. Sparse Fault-Tolerant BFS Trees. In Proceedings of the 21st European Symposium on Algorithms (ESA), pages 779-790, 2013. URL:
  28. Merav Parter and David Peleg. Sparse Fault-Tolerant BFS Structures. ACM Transaction on Algorithms, 13:11:1-11:24, 2016. URL:
  29. Ron Y. Pinter, Hadas Shachnai, and Meirav Zehavi. Deterministic Parameterized Algorithms for the Graph Motif Problem. In Proceedings of the 39th International Symposium on Mathematical Foundations of Computer Science (MFCS), pages 589-600, 2014. URL:
  30. Hanlin Ren. Improved Distance Sensitivity Oracles with Subcubic Preprocessing Time. Journal of Computer and System Sciences, 123:159-170, 2022. URL:
  31. Dekel Tsur. Faster Deterministic Parameterized Algorithm for k-Path. Theoretical Computer Science, 790:96-104, 2019. URL:
  32. Ryuhei Uehara and Yushi Uno. On Computing Longest Paths in Small Graph Classes. International Journal of Foundations of Computer Science, 18:911-930, 2007. URL:
  33. Jan van den Brand and Thatchaphol Saranurak. Sensitive Distance and Reachability Oracles for Large Batch Updates. In Proceedings of the 60th Symposium on Foundations of Computer Science (FOCS), pages 424-435, 2019. URL:
  34. Oren Weimann and Raphael Yuster. Replacement Paths and Distance Sensitivity Oracles via Fast Matrix Multiplication. ACM Transactions on Algorithms, 9:14:1-14:13, 2013. URL:
  35. R. Ryan Williams. Finding Paths of Length k in O^*(2^k) Time. Information Processing Letters, 109:315-318, 2009. URL: