Isolation Schemes for Problems on Decomposable Graphs

Authors Jesper Nederlof , Michał Pilipczuk , Céline M. F. Swennenhuis , Karol Węgrzycki

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Jesper Nederlof
  • Utrecht University, The Netherlands
Michał Pilipczuk
  • University of Warsaw, Poland
Céline M. F. Swennenhuis
  • Eindhoven University of Technology, The Netherlands
Karol Węgrzycki
  • Saarland University, Saarbrücken, Germany
  • Max Planck Institute for Informatics, Saarbrücken, Germany

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Jesper Nederlof, Michał Pilipczuk, Céline M. F. Swennenhuis, and Karol Węgrzycki. Isolation Schemes for Problems on Decomposable Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 50:1-50:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


The Isolation Lemma of Mulmuley, Vazirani and Vazirani [Combinatorica'87] provides a self-reduction scheme that allows one to assume that a given instance of a problem has a unique solution, provided a solution exists at all. Since its introduction, much effort has been dedicated towards derandomization of the Isolation Lemma for specific classes of problems. So far, the focus was mainly on problems solvable in polynomial time. In this paper, we study a setting that is more typical for NP-complete problems, and obtain partial derandomizations in the form of significantly decreasing the number of required random bits. In particular, motivated by the advances in parameterized algorithms, we focus on problems on decomposable graphs. For example, for the problem of detecting a Hamiltonian cycle, we build upon the rank-based approach from [Bodlaender et al., Inf. Comput.'15] and design isolation schemes that use - 𝒪(tlog n + log²{n}) random bits on graphs of treewidth at most t; - 𝒪(√n) random bits on planar or H-minor free graphs; and - 𝒪(n)-random bits on general graphs. In all these schemes, the weights are bounded exponentially in the number of random bits used. As a corollary, for every fixed H we obtain an algorithm for detecting a Hamiltonian cycle in an H-minor-free graph that runs in deterministic time 2^{𝒪(√n)} and uses polynomial space; this is the first algorithm to achieve such complexity guarantees. For problems of more local nature, such as finding an independent set of maximum size, we obtain isolation schemes on graphs of treedepth at most d that use 𝒪(d) random bits and assign polynomially-bounded weights. We also complement our findings with several unconditional and conditional lower bounds, which show that many of the results cannot be significantly improved.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Isolation Lemma
  • Derandomization
  • Hamiltonian Cycle
  • Exact Algorithms


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  1. Manindra Agrawal, Rohit Gurjar, and Thomas Thierauf. Impossibility of Derandomizing the Isolation Lemma for all Families. Electron. Colloquium Comput. Complex., 27:98, 2020. URL:
  2. Manindra Agrawal, Thanh Minh Hoang, and Thomas Thierauf. The Polynomially Bounded Perfect Matching Problem Is in NC². In STACS 2007, 24th Annual Symposium on Theoretical Aspects of Computer Science, pages 489-499, 2007. URL:
  3. Rahul Arora, Ashu Gupta, Rohit Gurjar, and Raghunath Tewari. Derandomizing Isolation Lemma for K_3, 3-free and K₅-free Bipartite Graphs. In 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016, pages 10:1-10:15, 2016. URL:
  4. Vikraman Arvind and Partha Mukhopadhyay. Derandomizing the isolation lemma and lower bounds for circuit size. In Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques, pages 276-289. Springer, 2008. Google Scholar
  5. Andreas Björklund. Determinant sums for undirected hamiltonicity. SIAM J. Comput., 43(1):280-299, 2014. URL:
  6. Hans L. Bodlaender, Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Inf. Comput., 243:86-111, 2015. URL:
  7. Chris Bourke, Raghunath Tewari, and N. V. Vinodchandran. Directed planar reachability is in unambiguous log-space. ACM Trans. Comput. Theory, 1(1):4:1-4:17, 2009. URL:
  8. Suresh Chari, Pankaj Rohatgi, and Aravind Srinivasan. Randomness-Optimal Unique Element Isolation with Applications to Perfect Matching and Related Problems. SIAM J. Comput., 24(5):1036-1050, 1995. URL:
  9. Jiehua Chen, Wojciech Czerwiński, Yann Disser, Andreas Emil Feldmann, Danny Hermelin, Wojciech Nadara, Michał Pilipczuk, Marcin Pilipczuk, Manuel Sorge, Bartlomiej Wróblewski, and Anna Zych-Pawlewicz. Efficient fully dynamic elimination forests with applications to detecting long paths and cycles. CoRR, abs/2006.00571, 2020. To appear in the proceedings of SODA 2021. URL:
  10. Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Fast Hamiltonicity Checking Via Bases of Perfect Matchings. J. ACM, 65(3):12:1-12:46, 2018. URL:
  11. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michał Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving Connectivity Problems Parameterized by Treewidth in Single Exponential Time. In 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, pages 150-159. IEEE, 2011. URL:
  12. Wojciech Czerwiński, Wojciech Nadara, and Marcin Pilipczuk. Improved Bounds for the Excluded-Minor Approximation of Treedepth. In 27th Annual European Symposium on Algorithms, ESA 2019, volume 144 of LIPIcs, pages 34:1-34:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL:
  13. Samir Datta, Raghav Kulkarni, and Sambuddha Roy. Deterministically isolating a perfect matching in bipartite planar graphs. Theory Comput. Syst., 47(3):737-757, 2010. URL:
  14. Zdenek Dvořák, Archontia C. Giannopoulou, and Dimitrios M. Thilikos. Forbidden graphs for tree-depth. Eur. J. Comb., 33(5):969-979, 2012. URL:
  15. Friedrich Eisenbrand, Christoph Hunkenschröder, Kim-Manuel Klein, Martin Koutecký, Asaf Levin, and Shmuel Onn. An algorithmic theory of integer programming. CoRR, abs/1904.01361, 2019. URL:
  16. Michael Elberfeld, Martin Grohe, and Till Tantau. Where first-order and monadic second-order logic coincide. In 27th Annual IEEE Symposium on Logic in Computer Science, LICS 2012, pages 265-274. IEEE Computer Society, 2012. URL:
  17. Stephen Fenner, Rohit Gurjar, and Thomas Thierauf. Bipartite perfect matching is in quasi-NC. SIAM Journal on Computing, pages STOC16-218, 2019. Google Scholar
  18. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Efficient computation of representative families with applications in parameterized and exact algorithms. J. ACM, 63(4):29:1-29:60, 2016. URL:
  19. Michael L. Fredman, János Komlós, and Endre Szemerédi. Storing a Sparse Table with 𝒪(1) Worst Case Access Time. J. ACM, 31(3):538-544, 1984. URL:
  20. Martin Fürer and Huiwen Yu. Space saving by dynamic algebraization based on tree-depth. Theory Comput. Syst., 61(2):283-304, 2017. URL:
  21. Dima Grigoriev and Marek Karpinski. The matching problem for bipartite graphs with polynomially bounded permanents is in NC (extended abstract). In 28th Annual Symposium on Foundations of Computer Science, Los Angeles, California, USA, 27-29 October 1987, pages 166-172. IEEE Computer Society, 1987. URL:
  22. Chetan Gupta, Vimal Raj Sharma, and Raghunath Tewari. Efficient Isolation of Perfect Matching in 𝒪(log n) Genus Bipartite Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020. Google Scholar
  23. Falko Hegerfeld and Stefan Kratsch. Solving Connectivity Problems Parameterized by Treedepth in Single-Exponential Time and Polynomial Space. In 37th International Symposium on Theoretical Aspects of Computer Science, STACS 2020, pages 29:1-29:16, 2020. URL:
  24. Bart M. P. Jansen and Jesper Nederlof. Computing the chromatic number using graph decompositions via matrix rank. Theor. Comput. Sci., 795:520-539, 2019. URL:
  25. Jason Li and Jesper Nederlof. Detecting Feedback Vertex Sets of Size k in 𝒪^⋆(2.7^k) Time. In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 971-989. SIAM, 2020. URL:
  26. Ketan Mulmuley, Umesh V. Vazirani, and Vijay V. Vazirani. Matching is as easy as matrix inversion. Comb., 7(1):105-113, 1987. URL:
  27. Jesper Nederlof, Michal Pilipczuk, Céline M. F. Swennenhuis, and Karol Wegrzycki. Isolation schemes for problems on decomposable graphs. CoRR, abs/2105.01465, 2021. URL:
  28. Jesper Nederlof, Michał Pilipczuk, Céline M. F. Swennenhuis, and Karol Węgrzycki. Hamiltonian cycle parameterized by treedepth in single exponential time and polynomial space. In 46th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2020, volume 12301 of Lecture Notes in Computer Science, pages 27-39. Springer, 2020. URL:
  29. Jaroslav Nešetřil and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012. URL:
  30. Michał Pilipczuk and Sebastian Siebertz. Polynomial bounds for centered colorings on proper minor-closed graph classes. In 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, pages 1501-1520. SIAM, 2019. URL:
  31. Michał Pilipczuk and Marcin Wrochna. On space efficiency of algorithms working on structural decompositions of graphs. ACM Trans. Comp. Theory, 9(4):18:1-18:36, 2018. URL:
  32. Klaus Reinhardt and Eric Allender. Making nondeterminism unambiguous. SIAM J. Comput., 29(4):1118-1131, 2000. URL:
  33. Barkley Rosser. Explicit bounds for some functions of prime numbers. American Journal of Mathematics, 63(1):211-232, 1941. URL:
  34. Ola Svensson and Jakub Tarnawski. The matching problem in general graphs is in Quasi-NC. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, pages 696-707, 2017. URL:
  35. Leslie G. Valiant and Vijay V. Vazirani. NP is as Easy as Detecting Unique Solutions. Theor. Comput. Sci., 47(3):85-93, 1986. URL:
  36. Dieter van Melkebeek and Gautam Prakriya. Derandomizing Isolation in Space-Bounded Settings. SIAM J. Comput., 48(3):979-1021, 2019. URL:
  37. Avi Wigderson. NL/poly ⊆ ⊕ L/poly (preliminary version). In Proceedings of the Ninth Annual Structure in Complexity Theory Conference, pages 59-62, 1994. URL:
  38. Ryan Williams. Finding paths of length k in 𝒪^⋆(2^k) time. Inf. Process. Lett., 109(6):315-318, 2009. URL:
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