Faster Exponential-Time Approximation Algorithms Using Approximate Monotone Local Search

Authors Barış Can Esmer , Ariel Kulik , Dániel Marx , Daniel Neuen , Roohani Sharma

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Barış Can Esmer
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
  • Saarbrücken Graduate School of Computer Science, Saarland Informatics Campus, Germany
Ariel Kulik
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Dániel Marx
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Daniel Neuen
  • School of Computing Science, Simon Fraser University, Burnaby, Canada
Roohani Sharma
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany

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Barış Can Esmer, Ariel Kulik, Dániel Marx, Daniel Neuen, and Roohani Sharma. Faster Exponential-Time Approximation Algorithms Using Approximate Monotone Local Search. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 50:1-50:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We generalize the monotone local search approach of Fomin, Gaspers, Lokshtanov and Saurabh [J.ACM 2019], by establishing a connection between parameterized approximation and exponential-time approximation algorithms for monotone subset minimization problems. In a monotone subset minimization problem the input implicitly describes a non-empty set family over a universe of size n which is closed under taking supersets. The task is to find a minimum cardinality set in this family. Broadly speaking, we use approximate monotone local search to show that a parameterized α-approximation algorithm that runs in c^k⋅n^𝒪(1) time, where k is the solution size, can be used to derive an α-approximation randomized algorithm that runs in dⁿ⋅n^𝒪(1) time, where d is the unique value in (1, 1+{c-1}/α) such that 𝒟(1/α‖{d-1}/{c-1}) = {ln c}/α and 𝒟(a‖b) is the Kullback-Leibler divergence. This running time matches that of Fomin et al. for α = 1, and is strictly better when α > 1, for any c > 1. Furthermore, we also show that this result can be derandomized at the expense of a sub-exponential multiplicative factor in the running time. We use an approximate variant of the exhaustive search as a benchmark for our algorithm. We show that the classic 2ⁿ⋅n^𝒪(1) exhaustive search can be adapted to an α-approximate exhaustive search that runs in time (1+exp(-α⋅ℋ(1/(α))))ⁿ⋅n^𝒪(1), where ℋ is the entropy function. Furthermore, we provide a lower bound stating that the running time of this α-approximate exhaustive search is the best achievable running time in an oracle model. When compared to approximate exhaustive search, and to other techniques, the running times obtained by approximate monotone local search are strictly better for any α ≥ 1, c > 1. We demonstrate the potential of approximate monotone local search by deriving new and faster exponential approximation algorithms for Vertex Cover, 3-Hitting Set, Directed Feedback Vertex Set, Directed Subset Feedback Vertex Set, Directed Odd Cycle Transversal and Undirected Multicut. For instance, we get a 1.1-approximation algorithm for Vertex Cover with running time 1.114ⁿ⋅n^𝒪(1), improving upon the previously best known 1.1-approximation running in time 1.127ⁿ⋅n^𝒪(1) by Bourgeois et al. [DAM 2011].

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Mathematics of computing → Approximation algorithms
  • Mathematics of computing → Probabilistic algorithms
  • parameterized approximations
  • exponential approximations
  • monotone local search


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  1. Sanjeev Arora, Boaz Barak, and David Steurer. Subexponential algorithms for unique games and related problems. J. ACM, 62(5):42:1-42:25, 2015. URL:
  2. Nikhil Bansal, Parinya Chalermsook, Bundit Laekhanukit, Danupon Nanongkai, and Jesper Nederlof. New tools and connections for exponential-time approximation. Algorithmica, 81(10):3993-4009, 2019. URL:
  3. Nicolas Bourgeois, Bruno Escoffier, and Vangelis Th. Paschos. Approximation of max independent set, min vertex cover and related problems by moderately exponential algorithms. Discret. Appl. Math., 159(17):1954-1970, 2011. URL:
  4. Ljiljana Brankovic and Henning Fernau. Parameterized approximation algorithms for hitting set. In Roberto Solis-Oba and Giuseppe Persiano, editors, Approximation and Online Algorithms - 9th International Workshop, WAOA 2011, Saarbrücken, Germany, September 8-9, 2011, Revised Selected Papers, volume 7164 of Lecture Notes in Computer Science, pages 63-76. Springer, 2011. URL:
  5. Ljiljana Brankovic and Henning Fernau. A novel parameterised approximation algorithm for minimum vertex cover. Theor. Comput. Sci., 511:85-108, 2013. URL:
  6. Thomas M. Cover and Joy A. Thomas. Elements of information theory. Wiley-Interscience, 2nd edition, 2006. URL:
  7. Marek Cygan, Lukasz Kowalik, and Mateusz Wykurz. Exponential-time approximation of weighted set cover. Inf. Process. Lett., 109(16):957-961, 2009. URL:
  8. Pavel Dvorák, Andreas Emil Feldmann, Dusan Knop, Tomás Masarík, Tomás Toufar, and Pavel Veselý. Parameterized approximation schemes for Steiner trees with small number of Steiner vertices. SIAM J. Discret. Math., 35(1):546-574, 2021. URL:
  9. Barış Can Esmer, Ariel Kulik, Dániel Marx, Daniel Neuen, and Roohani Sharma. Faster exponential-time approximation algorithms using approximate monotone local search. CoRR, abs/2206.13481, 2022. URL:
  10. Uriel Feige and Mohammad Mahdian. Finding small balanced separators. In Jon M. Kleinberg, editor, Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, May 21-23, 2006, pages 375-384. ACM, 2006. URL:
  11. Andreas Emil Feldmann, Karthik C. S., Euiwoong Lee, and Pasin Manurangsi. A survey on approximation in parameterized complexity: Hardness and algorithms. Algorithms, 13(6):146, 2020. URL:
  12. Michael R. Fellows, Ariel Kulik, Frances A. Rosamond, and Hadas Shachnai. Parameterized approximation via fidelity preserving transformations. J. Comput. Syst. Sci., 93:30-40, 2018. URL:
  13. Fedor V. Fomin, Serge Gaspers, Daniel Lokshtanov, and Saket Saurabh. Exact algorithms via monotone local search. J. ACM, 66(2):8:1-8:23, 2019. URL:
  14. Anupam Gupta, Euiwoong Lee, and Jason Li. Faster exact and approximate algorithms for k-cut. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 113-123. IEEE Computer Society, 2018. URL:
  15. Anupam Gupta, Euiwoong Lee, and Jason Li. An FPT algorithm beating 2-approximation for k-cut. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 2821-2837. SIAM, 2018. URL:
  16. Ken-ichi Kawarabayashi and Bingkai Lin. A nearly 5/3-approximation FPT algorithm for min-k-cut. 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 990-999. SIAM, 2020. URL:
  17. Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2-ε. J. Comput. Syst. Sci., 74(3):335-349, 2008. URL:
  18. Ariel Kulik and Hadas Shachnai. Analysis of two-variable recurrence relations with application to parameterized approximations. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 762-773. IEEE, 2020. URL:
  19. Nikolai N. Kuzjurin. Explicit constructions of Rödl’s asymptotically good packings and coverings. Comb. Probab. Comput., 9(3):265-276, 2000. URL:
  20. Edward Lee. Exponential time algorithms via separators and random subsets. PhD thesis, University of New South Wales, 2021. URL:
  21. Euiwoong Lee. Partitioning a graph into small pieces with applications to path transversal. Math. Program., 177(1-2):1-19, 2019. URL:
  22. Daniel Lokshtanov, Pranabendu Misra, M. S. Ramanujan, Saket Saurabh, and Meirav Zehavi. FPT-approximation for FPT problems. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10-13, 2021, pages 199-218. SIAM, 2021. URL:
  23. Daniel Lokshtanov, Saket Saurabh, and Vaishali Surianarayanan. A parameterized approximation scheme for min k-cut. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 798-809. IEEE, 2020. URL:
  24. Pasin Manurangsi. A note on max k-vertex cover: Faster fpt-as, smaller approximate kernel and improved approximation. In Jeremy T. Fineman and Michael Mitzenmacher, editors, 2nd Symposium on Simplicity in Algorithms, SOSA 2019, January 8-9, 2019, San Diego, CA, USA, volume 69 of OASIcs, pages 15:1-15:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL:
  25. Pasin Manurangsi and Luca Trevisan. Mildly exponential time approximation algorithms for vertex cover, balanced separator and uniform sparsest cut. In Eric Blais, Klaus Jansen, José D. P. Rolim, and David Steurer, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2018, August 20-22, 2018 - Princeton, NJ, USA, volume 116 of LIPIcs, pages 20:1-20:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL:
  26. Dániel Marx. Parameterized complexity and approximation algorithms. Comput. J., 51(1):60-78, 2008. URL:
  27. Dániel Marx and Igor Razgon. Constant ratio fixed-parameter approximation of the edge multicut problem. Inf. Process. Lett., 109(20):1161-1166, 2009. URL:
  28. Igor Razgon. Computing minimum directed feedback vertex set in O*(1.9977ⁿ). In Giuseppe F. Italiano, Eugenio Moggi, and Luigi Laura, editors, Theoretical Computer Science, 10th Italian Conference, ICTCS 2007, Rome, Italy, October 3-5, 2007, Proceedings, pages 70-81. World Scientific, 2007. URL:
  29. Piotr Skowron and Piotr Faliszewski. Chamberlin-Courant rule with approval ballots: approximating the MaxCover problem with bounded frequencies in FPT time. J. Artificial Intelligence Res., 60:687-716, 2017. URL:
  30. Vijay V. Vazirani. Approximation algorithms. Springer, 2001. URL:
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