Time-Approximation Trade-offs for Inapproximable Problems

Authors Édouard Bonnet, Michael Lampis, Vangelis Th. Paschos

Thumbnail PDF


  • Filesize: 0.68 MB
  • 14 pages

Document Identifiers

Author Details

Édouard Bonnet
Michael Lampis
Vangelis Th. Paschos

Cite AsGet BibTex

Édouard Bonnet, Michael Lampis, and Vangelis Th. Paschos. Time-Approximation Trade-offs for Inapproximable Problems. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


In this paper we focus on problems which do not admit a constant-factor approximation in polynomial time and explore how quickly their approximability improves as the allowed running time is gradually increased from polynomial to (sub-)exponential. We tackle a number of problems: For MIN INDEPENDENT DOMINATING SET, MAX INDUCED PATH, FOREST and TREE, for any r(n), a simple, known scheme gives an approximation ratio of r in time roughly r^{n/r}. We show that, for most values of r, if this running time could be significantly improved the ETH would fail. For MAX MINIMAL VERTEX COVER we give a non-trivial sqrt{r}-approximation in time 2^{n/{r}}. We match this with a similarly tight result. We also give a log(r)-approximation for MIN ATSP in time 2^{n/r} and an r-approximation for MAX GRUNDY COLORING in time r^{n/r}. Furthermore, we show that MIN SET COVER exhibits a curious behavior in this super-polynomial setting: for any delta>0 it admits an m^delta-approximation, where m is the number of sets, in just quasi-polynomial time. We observe that if such ratios could be achieved in polynomial time, the ETH or the Projection Games Conjecture would fail.
  • Algorithm
  • Complexity
  • Polynomial and Subexponential Approximation
  • Reduction
  • Inapproximability


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


  1. Arash Asadpour, Michel X. Goemans, Aleksander Madry, Shayan Oveis Gharan, and Amin Saberi. An O(log n/ log log n)-approximation algorithm for the asymmetric traveling salesman problem. In Moses Charikar, editor, Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'10, pages 379-389. SIAM, 2010. URL: http://dx.doi.org/10.1137/1.9781611973075.32.
  2. Nicolas Bourgeois, Federico Della Croce, Bruno Escoffier, and Vangelis Th. Paschos. Fast algorithms for min independent dominating set. Discrete Applied Mathematics, 161(4-5):558-572, 2013. URL: http://dx.doi.org/10.1016/j.dam.2012.01.003.
  3. Nicolas Bourgeois, Bruno Escoffier, and Vangelis Th. Paschos. Approximation of min coloring by moderately exponential algorithms. Information Processing Letters, 109(16):950-954, 2009. URL: http://dx.doi.org/10.1016/j.ipl.2009.05.002.
  4. Nicolas Bourgeois, Bruno Escoffier, and Vangelis Th. Paschos. Efficient approximation of min set cover by moderately exponential algorithms. Theoretical Computer Science, 410(21-23):2184-2195, 2009. URL: http://dx.doi.org/10.1016/j.tcs.2009.02.007.
  5. Nicolas Bourgeois, Bruno Escoffier, and Vangelis Th. Paschos. Approximation of max independent set, min vertex cover and related problems by moderately exponential algorithms. Discrete Applied Mathematics, 159(17):1954-1970, 2011. URL: http://dx.doi.org/10.1016/j.dam.2011.07.009.
  6. Parinya Chalermsook, Bundit Laekhanukit, and Danupon Nanongkai. Independent set, induced matching, and pricing: Connections and tight (subexponential time) approximation hardnesses. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS'13, pages 370-379, 2013. URL: http://dx.doi.org/10.1109/FOCS.2013.47.
  7. Rajesh Hemant Chitnis, MohammadTaghi Hajiaghayi, and Guy Kortsarz. Fixed-parameter and approximation algorithms: A new look. In G. Gutin and S. Szeider, editors, Parameterized and Exact Computation - 8th International Symposium, IPEC'13, Revised Selected Papers, volume 8246 of Lecture Notes in Computer Science, pages 110-122. Springer, 2013. URL: http://dx.doi.org/10.1007/978-3-319-03898-8_11.
  8. Marek Cygan, Lukasz Kowalik, and Mateusz Wykurz. Exponential-time approximation of weighted set cover. Information Processing Letters, 109(16):957-961, 2009. URL: http://dx.doi.org/10.1016/j.ipl.2009.05.003.
  9. Marek Cygan and Marcin Pilipczuk. Exact and approximate bandwidth. Theoretical Computer Science, 411(40-42):3701-3713, 2010. URL: http://dx.doi.org/10.1016/j.tcs.2010.06.018.
  10. Marek Cygan, Marcin Pilipczuk, and Jakub Onufry Wojtaszczyk. Capacitated domination faster than O(2ⁿ). Information Processing Letters, 111(23-24):1099-1103, 2011. URL: http://dx.doi.org/10.1016/j.ipl.2011.09.004.
  11. Irit Dinur. The PCP theorem by gap amplification. Journal of the ACM, 54(3), 2007. Article 12. URL: http://dx.doi.org/10.1145/1236457.1236459.
  12. Irit Dinur and David Steurer. Analytical approach to parallel repetition. In D. B. Shmoys, editor, Symposium on Theory of Computing, STOC'14, pages 624-633. ACM, 2014. URL: http://dl.acm.org/citation.cfm?id=2591796, URL: http://dx.doi.org/10.1145/2591796.2591884.
  13. Rodney G. Downey, Michael R. Fellows, Catherine McCartin, and Frances A. Rosamond. Parameterized approximation of dominating set problems. Information Processing Letters, 109(1):68-70, 2008. URL: http://dx.doi.org/10.1016/j.ipl.2008.09.017.
  14. Alan M. Frieze, Giulia Galbiati, and Francesco Maffioli. On the worst-case performance of some algorithms for the asymmetric traveling salesman problem. Networks, 12(1):23-39, 1982. URL: http://dx.doi.org/10.1002/net.3230120103.
  15. Magnús M. Halldórsson. Approximating the minimum maximal independence number. Information Processing Letters, 46(4):169-172, 1993. URL: http://dx.doi.org/10.1016/0020-0190(93)90022-2.
  16. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computers and System Sciences, 63(4):512-530, 2001. URL: http://dx.doi.org/10.1006/jcss.2001.1774.
  17. Viggo Kann. Strong lower bounds on the approximability of some NPO pb-complete maximization problems. In J. Wiedermann and P. Hájek, editors, Mathematical Foundations of Computer Science 1995, 20th International Symposium, MFCS'95, volume 969 of Lecture Notes in Computer Science, pages 227-236. Springer, 1995. URL: http://dx.doi.org/10.1007/3-540-60246-1_129.
  18. Marek Karpinski, Michael Lampis, and Richard Schmied. New inapproximability bounds for TSP. In L. Cai, S.-W. Cheng, and T. W. Lam, editors, Algorithms and Computation - 24th International Symposium, ISAAC'13, volume 8283 of Lecture Notes in Computer Science, pages 568-578. Springer, 2013. URL: http://dx.doi.org/10.1007/978-3-642-45030-3_53.
  19. Guy Kortsarz. A lower bound for approximating grundy numbering. Discrete Mathematics & Theoretical Computer Science, 9(1), 2007. Google Scholar
  20. Carsten Lund and Mihalis Yannakakis. The approximation of maximum subgraph problems. In A. Lingas, R. G. Karlsson, and S. Carlsson, editors, Automata, Languages and Programming, 20nd International Colloquium, ICALP'93, volume 700 of Lecture Notes in Computer Science, pages 40-51. Springer, 1993. URL: http://dx.doi.org/10.1007/3-540-56939-1_60.
  21. Dana Moshkovitz. The projection games conjecture and the NP-hardness of ln n-approximating set-cover. In A. Gupta, K. Jansen, J.D.P Rolim, and R. Servedio, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 15th International Workshop, APPROX'12, and 16th International Workshop, RANDOM'12, volume 7408 of Lecture Notes in Computer Science, pages 276-287. Springer, 2012. URL: http://dx.doi.org/10.1007/978-3-642-32512-0_24.
  22. Dana Moshkovitz and Ran Raz. Two-query PCP with subconstant error. Journal of the ACM, 57(5), 2010. Article 29. URL: http://dx.doi.org/10.1145/1754399.1754402.
  23. Jelani Nelson. A note on set cover inapproximability independent of universe size. Electronic Colloquium on Computational Complexity (ECCC), 14(105), 2007. Google Scholar