Approximate Monotone Local Search for Weighted Problems

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

Thumbnail PDF


  • Filesize: 0.85 MB
  • 23 pages

Document Identifiers

Author Details

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
  • University of Bremen, Germany
Roohani Sharma
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany

Cite AsGet BibTex

Barış Can Esmer, Ariel Kulik, Dániel Marx, Daniel Neuen, and Roohani Sharma. Approximate Monotone Local Search for Weighted Problems. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 17:1-17:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


In a recent work, Esmer et al. describe a simple method - Approximate Monotone Local Search - to obtain exponential approximation algorithms from existing parameterized exact algorithms, polynomial-time approximation algorithms and, more generally, parameterized approximation algorithms. In this work, we generalize those results to the weighted setting. More formally, we consider monotone subset minimization problems over a weighted universe of size n (e.g., Vertex Cover, d-Hitting Set and Feedback Vertex Set). We consider a model where the algorithm is only given access to a subroutine that finds a solution of weight at most α ⋅ W (and of arbitrary cardinality) in time c^k ⋅ n^{𝒪(1)} where W is the minimum weight of a solution of cardinality at most k. In the unweighted setting, Esmer et al. determine the smallest value d for which a β-approximation algorithm running in time dⁿ ⋅ n^{𝒪(1)} can be obtained in this model. We show that the same dependencies also hold in a weighted setting in this model: for every fixed ε > 0 we obtain a β-approximation algorithm running in time 𝒪((d+ε)ⁿ), for the same d as in the unweighted setting. Similarly, we also extend a β-approximate brute-force search (in a model which only provides access to a membership oracle) to the weighted setting. Using existing approximation algorithms and exact parameterized algorithms for weighted problems, we obtain the first exponential-time β-approximation algorithms that are better than brute force for a variety of problems including Weighted Vertex Cover, Weighted d-Hitting Set, Weighted Feedback Vertex Set and Weighted Multicut.

Subject Classification

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


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Akanksha Agrawal, Sudeshna Kolay, Daniel Lokshtanov, and Saket Saurabh. A faster FPT algorithm and a smaller kernel for block graph vertex deletion. In Evangelos Kranakis, Gonzalo Navarro, and Edgar Chávez, editors, LATIN 2016: Theoretical Informatics - 12th Latin American Symposium, Ensenada, Mexico, April 11-15, 2016, Proceedings, volume 9644 of Lecture Notes in Computer Science, pages 1-13. Springer, 2016. URL:
  2. 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:
  3. Vineet Bafna, Piotr Berman, and Toshihiro Fujito. A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM J. Discret. Math., 12(3):289-297, 1999. URL:
  4. 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:
  5. Reuven Bar-Yehuda and Shimon Even. A linear-time approximation algorithm for the weighted vertex cover problem. J. Algorithms, 2(2):198-203, 1981. URL:
  6. 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:
  7. Nader H. Bshouty and Lynn Burroughs. Massaging a linear programming solution to give a 2-approximation for a generalization of the vertex cover problem. In Michel Morvan, Christoph Meinel, and Daniel Krob, editors, STACS 98, 15th Annual Symposium on Theoretical Aspects of Computer Science, Paris, France, February 25-27, 1998, Proceedings, volume 1373 of Lecture Notes in Computer Science, pages 298-308. Springer, 1998. URL:
  8. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL:
  9. Marek Cygan, Lukasz Kowalik, and Mateusz Wykurz. Exponential-time approximation of weighted set cover. Inf. Process. Lett., 109(16):957-961, 2009. URL:
  10. Bruno Escoffier, Vangelis Th. Paschos, and Emeric Tourniaire. Super-polynomial approximation branching algorithms. RAIRO Oper. Res., 50(4-5):979-994, 2016. URL:
  11. Barış Can Esmer, Ariel Kulik, Dániel Marx, Daniel Neuen, and Roohani Sharma. Faster exponential-time approximation algorithms using approximate monotone local search. In Shiri Chechik, Gonzalo Navarro, Eva Rotenberg, and Grzegorz Herman, editors, 30th Annual European Symposium on Algorithms, ESA 2022, September 5-9, 2022, Berlin/Potsdam, Germany, volume 244 of LIPIcs, pages 50:1-50:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL:
  12. Barış Can Esmer, Ariel Kulik, Dániel Marx, Daniel Neuen, and Roohani Sharma. Optimally repurposing existing algorithms to obtain exponential-time approximations. CoRR, abs/2306.15331, 2023. To be published at SODA 2024., URL:
  13. Guy Even, Joseph Naor, and Leonid Zosin. An 8-approximation algorithm for the subset feedback vertex set problem. SIAM J. Comput., 30(4):1231-1252, 2000. URL:
  14. Fedor V. Fomin, Serge Gaspers, Dieter Kratsch, Mathieu Liedloff, and Saket Saurabh. Iterative compression and exact algorithms. Theor. Comput. Sci., 411(7-9):1045-1053, 2010. URL:
  15. 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:
  16. Fedor V. Fomin and Dieter Kratsch. Exact Exponential Algorithms. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2010. URL:
  17. 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:
  18. 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:
  19. 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:
  20. Rolf Niedermeier and Peter Rossmanith. On efficient fixed-parameter algorithms for weighted vertex cover. J. Algorithms, 47(2):63-77, 2003. URL:
  21. Hadas Shachnai and Meirav Zehavi. A multivariate framework for weighted FPT algorithms. J. Comput. Syst. Sci., 89:157-189, 2017. URL:
  22. Magnus Wahlström. A tighter bound for counting max-weight solutions to 2sat instances. In Martin Grohe and Rolf Niedermeier, editors, Parameterized and Exact Computation, Third International Workshop, IWPEC 2008, Victoria, Canada, May 14-16, 2008. Proceedings, volume 5018 of Lecture Notes in Computer Science, pages 202-213. Springer, 2008. URL: