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Towards Hardness of Approximation for Polynomial Time Problems

Authors Amir Abboud, Arturs Backurs

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Amir Abboud
Arturs Backurs

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Amir Abboud and Arturs Backurs. Towards Hardness of Approximation for Polynomial Time Problems. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 11:1-11:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.ITCS.2017.11

Abstract

Proving hardness of approximation is a major challenge in the field of fine-grained complexity and conditional lower bounds in P.
How well can the Longest Common Subsequence (LCS) or the Edit Distance be approximated by an algorithm that runs in near-linear time?
In this paper, we make progress towards answering these questions.
We introduce a framework that exhibits barriers for truly subquadratic and deterministic algorithms with good approximation guarantees.
Our framework highlights a novel connection between deterministic approximation algorithms for natural problems in P and circuit lower bounds.

In particular, we discover a curious connection of the following form:
if there exists a \delta>0 such that for all \eps>0 there is a deterministic (1+\eps)-approximation algorithm for LCS on two sequences of length n over an alphabet of size n^{o(1)} that runs in O(n^{2-\delta}) time, then a certain plausible hypothesis is refuted, and the class E^NP does not have non-uniform linear size Valiant Series-Parallel circuits.
Thus, designing a "truly subquadratic PTAS" for LCS is as hard as resolving an old open question in complexity theory.

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Keywords
  • LCS
  • Edit Distance
  • Hardness in P

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References

  1. Amir Abboud, Arturs Backurs, Thomas Dueholm Hansen, Virginia Vassilevska Williams, and Or Zamir. Subtree isomorphism revisited. In Proc. of 27th SODA, pages 1256-1271, 2016. Google Scholar
  2. Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. Tight Hardness Results for LCS and other Sequence Similarity Measures. In Proc. of 56th FOCS, pages 59-78, 2015. Google Scholar
  3. Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. If the current clique algorithms are optimal, so is valiant’s parser. In Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium on, pages 98-117. IEEE, 2015. Google Scholar
  4. Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, APSP and diameter. In Proc. of 26th SODA, pages 1681-1697, 2015. Google Scholar
  5. Amir Abboud, Thomas Dueholm Hansen, Virginia Vassilevska Williams, and Ryan Williams. Simulating Branching Programs with Edit Distance and Friends or: A Polylog Shaved is a Lower Bound Made. In STOC'16, 2016. Google Scholar
  6. Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In Proc. of 55th FOCS, pages 434-443, 2014. Google Scholar
  7. Amir Abboud, Virginia Vassilevska Williams, and Joshua R. Wang. Approximation and fixed parameter subquadratic algorithms for radius and diameter in sparse graphs. In Proc. of 27th SODA, pages 377-391, 2016. Google Scholar
  8. Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann. Consequences of faster sequence alignment. In Proc. of 41st ICALP, pages 39-51, 2014. Google Scholar
  9. Amir Abboud, Virginia Vassilevska Williams, and Huacheng Yu. Matching triangles and basing hardness on an extremely popular conjecture. In Proc. of 47th STOC, pages 41-50, 2015. Google Scholar
  10. Amir Abboud, Ryan Williams, and Huacheng Yu. More applications of the polynomial method to algorithm design. In Proc. of 26th SODA, pages 218-230, 2015. Google Scholar
  11. Amir Abboud, Virginia Vassilevska Williams, and Joshua Wang. Approximation and fixed parameter subquadratic algorithms for radius and diameter. arXiv preprint arXiv:1506.01799, 2015. Google Scholar
  12. Josh Alman, Timothy M Chan, and Ryan Williams. Polynomial representations of threshold functions and algorithmic applications. In to appear at FOCS, 2016. Google Scholar
  13. Stephen F Altschul, Thomas L Madden, Alejandro A Schäffer, Jinghui Zhang, Zheng Zhang, Webb Miller, and David J Lipman. Gapped blast and psi-blast: a new generation of protein database search programs. Nucleic acids research, 25(17):3389-3402, 1997. Google Scholar
  14. A. Amir, T. M. Chan, M. Lewenstein, and N. Lewenstein. On hardness of jumbled indexing. In Proc. ICALP, volume 8572, pages 114-125, 2014. Google Scholar
  15. Alexandr Andoni, Robert Krauthgamer, and Krzysztof Onak. Polylogarithmic approximation for edit distance and the asymmetric query complexity. In FOCS, pages 377-386, 2010. Google Scholar
  16. László Babai, Lance Fortnow, Noam Nisan, and Avi Wigderson. Bpp has subexponential time simulations unless exptime has publishable proofs. In Structure in Complexity Theory Conference, 1991., Proceedings of the Sixth Annual, pages 213-219. IEEE, 1991. Google Scholar
  17. Rolf Backofen, Dekel Tsur, Shay Zakov, and Michal Ziv-Ukelson. Sparse rna folding: Time and space efficient algorithms. Journal of Discrete Algorithms, 9(1):12-31, 2011. Google Scholar
  18. Arturs Backurs, Nishanth Dikkala, and Christos Tzamos. Tight hardness results for maximum weight rectangles. arXiv preprint arXiv:1602.05837, 2016. Google Scholar
  19. Arturs Backurs and Piotr Indyk. Edit Distance Cannot Be Computed in Strongly Subquadratic Time (unless SETH is false). In Proc. of 47th STOC, pages 51-58, 2015. Google Scholar
  20. Arturs Backurs and Piotr Indyk. Which regular expression patterns are hard to match? arXiv preprint arXiv:1511.07070, 2015. Google Scholar
  21. Ziv Bar-Yossef, TS Jayram, Robert Krauthgamer, and Ravi Kumar. Approximating edit distance efficiently. In Foundations of Computer Science, 2004. Proceedings. 45th Annual IEEE Symposium on, pages 550-559. IEEE, 2004. Google Scholar
  22. Tuğkan Batu, Funda Ergun, and Cenk Sahinalp. Oblivious string embeddings and edit distance approximations. In Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, pages 792-801. Society for Industrial and Applied Mathematics, 2006. Google Scholar
  23. Eli Ben-Sasson, Oded Goldreich, Prahladh Harsha, Madhu Sudan, and Salil Vadhan. Short pcps verifiable in polylogarithmic time. In Computational Complexity, 2005. Proceedings. Twentieth Annual IEEE Conference on, pages 120-134. IEEE, 2005. Google Scholar
  24. Eli Ben-Sasson, Oded Goldreich, Prahladh Harsha, Madhu Sudan, and Salil P. Vadhan. Robust pcps of proximity, shorter pcps and applications to coding. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, USA, June 13-16, 2004, pages 1-10, 2004. URL: http://dx.doi.org/10.1145/1007352.1007361.
  25. Eli Ben-Sasson and Madhu Sudan. Short pcps with polylog query complexity. SIAM Journal on Computing, 38(2):551-607, 2008. Google Scholar
  26. Eli Ben-Sasson and Emanuele Viola. Short pcps with projection queries. In ICALP, Part I, pages 163-173, 2014. Google Scholar
  27. Lasse Bergroth, Harri Hakonen, and Timo Raita. New approximation algorithms for longest common subsequences. In String Processing and Information Retrieval: A South American Symposium, 1998. Proceedings, pages 32-40. IEEE, 1998. Google Scholar
  28. Lasse Bergroth, Harri Hakonen, and Timo Raita. A survey of longest common subsequence algorithms. In String Processing and Information Retrieval, 2000. SPIRE 2000. Proceedings. Seventh International Symposium on, pages 39-48. IEEE, 2000. Google Scholar
  29. Mark Braverman, Young Kun-Ko, and Omri Weinstein. Approximating the best nash equilibrium in n^o^(log n)-time breaks the exponential time hypothesis. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 970-982, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.66.
  30. Karl Bringmann. Why walking the dog takes time: Frechet distance has no strongly subquadratic algorithms unless seth fails. In Proc. of 55th FOCS, pages 661-670, 2014. Google Scholar
  31. Karl Bringmann and Marvin Künnemann. Improved approximation for fréchet distance on c-packed curves matching conditional lower bounds. CoRR, abs/1408.1340, 2014. URL: http://arxiv.org/abs/1408.1340.
  32. Karl Bringmann and Marvin Kunnemann. Quadratic Conditional Lower Bounds for String Problems and Dynamic Time Warping. In Proc. of 56th FOCS, pages 79-97, 2015. Google Scholar
  33. Karl Bringmann and Wolfgang Mulzer. Approximability of the Discrete Fréchet Distance. In Proc. of 31st SoCG, pages 739-753, 2015. Google Scholar
  34. Chris Calabro. A lower bound on the size of series-parallel graphs dense in long paths. In Electronic Colloquium on Computational Complexity (ECCC), volume 15, 2008. Google Scholar
  35. Chris Calabro, Russell Impagliazzo, and Ramamohan Paturi. The complexity of satisfiability of small depth circuits. In Proc. of 4th IWPEC, pages 75-85, 2009. Google Scholar
  36. Marco L Carmosino, Jiawei Gao, Russell Impagliazzo, Ivan Mihajlin, Ramamohan Paturi, and Stefan Schneider. Nondeterministic extensions of the strong exponential time hypothesis and consequences for non-reducibility. In Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, pages 261-270. ACM, 2016. Google Scholar
  37. Marco L Carmosino, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. Tighter connections between derandomization and circuit lower bounds. In LIPIcs-Leibniz International Proceedings in Informatics, volume 40. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2015. Google Scholar
  38. Diptarka Chakraborty, Elazar Goldenberg, and Michal Kouckỳ. Streaming algorithms for embedding and computing edit distance in the low distance regime. In Proceedings of the Forty-eighth Annual ACM Symposium on Theory of Computing, 2015. Google Scholar
  39. Timothy M Chan and Ryan Williams. Deterministic apsp, orthogonal vectors, and more: Quickly derandomizing razborov-smolensky. In Proceedings of the Twenty-seventh Annual ACM-SIAM Symposium on Discrete Algorithms. SODA, 2016. Google Scholar
  40. Yi-Jun Chang. Hardness of rna folding problem with four symbols. arXiv preprint arXiv:1511.04731, 2015. Google Scholar
  41. Krishnendu Chatterjee, Wolfgang Dvořák, Monika Henzinger, and Veronika Loitzenbauer. Model and objective separation with conditional lower bounds: Disjunction is harder than conjunction. arXiv preprint arXiv:1602.02670, 2016. Google Scholar
  42. Shiri Chechik, Daniel H Larkin, Liam Roditty, Grant Schoenebeck, Robert E Tarjan, and Virginia Vassilevska Williams. Better approximation algorithms for the graph diameter. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1041-1052. SIAM, 2014. Google Scholar
  43. F Chin and Chung Keung Poon. Performance analysis of some simple heuristics for computing longest common subsequences. Algorithmica, 12(4-5):293-311, 1994. Google Scholar
  44. Thomas H.. Cormen, Charles Eric Leiserson, Ronald L Rivest, and Clifford Stein. Introduction to algorithms, volume 6. MIT press Cambridge, 2001. Google Scholar
  45. Maxime Crochemore, Costas S Iliopoulos, Yoan J Pinzon, and James F Reid. A fast and practical bit-vector algorithm for the longest common subsequence problem. Information Processing Letters, 80(6):279-285, 2001. Google Scholar
  46. Marek Cygan, Holger Dell, Daniel Lokshtanov, Dániel Marx, Jesper Nederlof, Yoshio Okamoto, Ramamohan Paturi, Saket Saurabh, and Magnus Wahlström. On problems as hard as cnf-sat. In Computational Complexity (CCC), 2012 IEEE 27th Annual Conference on, pages 74-84. IEEE, 2012. Google Scholar
  47. Søren Dahlgaard. On the hardness of partially dynamic graph problems and connections to diameter. arXiv preprint arXiv:1602.06705, 2016. Google Scholar
  48. J Boutet de Monvel. Extensive simulations for longest common subsequences. The European Physical Journal B-Condensed Matter and Complex Systems, 7(2):293-308, 1999. Google Scholar
  49. Robert C Edgar and Serafim Batzoglou. Multiple sequence alignment. Current opinion in structural biology, 16(3):368-373, 2006. Google Scholar
  50. Lance Fortnow and Adam R Klivans. Efficient learning algorithms yield circuit lower bounds. Journal of Computer and System Sciences, 75(1):27-36, 2009. Google Scholar
  51. Yelena Frid and Dan Gusfield. A simple, practical and complete o-time algorithm for rna folding using the four-russians speedup. Algorithms for Molecular Biology, 5(1):1, 2010. Google Scholar
  52. A. Gajentaan and M. H. Overmars. On a class of o(n²) problems in computational geometry. Comput. Geom. Theory Appl., 45(4):140-152, 2012. Google Scholar
  53. Ofer Grossman and Dana Moshkovitz. Amplification and derandomization without slowdown. arXiv preprint arXiv:1509.08123, 2015. Google Scholar
  54. Dan Gusfield. Algorithms on strings, trees and sequences: computer science and computational biology. Cambridge university press, 1997. Google Scholar
  55. Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak. Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, pages 21-30. ACM, 2015. Google Scholar
  56. Russell Impagliazzo, Valentine Kabanets, and Avi Wigderson. In search of an easy witness: Exponential time vs. probabilistic polynomial time. Journal of Computer and System Sciences, 65(4):672-694, 2002. Google Scholar
  57. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. Journal of Computer and System Sciences, 62(2):367-375, 2001. Google Scholar
  58. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63:512-530, 2001. Google Scholar
  59. Russell Impagliazzo and Avi Wigderson. P= bpp if e requires exponential circuits: Derandomizing the xor lemma. In Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, pages 220-229. ACM, 1997. Google Scholar
  60. Hamid Jahanjou, Eric Miles, and Emanuele Viola. Local reductions. In Automata, Languages, and Programming, pages 749-760. Springer, 2015. Google Scholar
  61. Valentine Kabanets and Russell Impagliazzo. Derandomizing polynomial identity tests means proving circuit lower bounds. Computational Complexity, 13(1-2):1-46, 2004. Google Scholar
  62. Richard M Karp and Richard Lipton. Turing machines that take advice. Enseign. Math, 28(2):191-209, 1982. Google Scholar
  63. Kazutaka Katoh and Daron M Standley. Mafft multiple sequence alignment software version 7: improvements in performance and usability. Molecular biology and evolution, 30(4):772-780, 2013. Google Scholar
  64. Adam Klivans, Pravesh Kothari, and Igor C Oliveira. Constructing hard functions using learning algorithms. In Computational Complexity (CCC), 2013 IEEE Conference on, pages 86-97. IEEE, 2013. Google Scholar
  65. Gad M Landau, Eugene W Myers, and Jeanette P Schmidt. Incremental string comparison. SIAM Journal on Computing, 27(2):557-582, 1998. Google Scholar
  66. William J Masek and Michael S Paterson. A faster algorithm computing string edit distances. Journal of Computer and System sciences, 20(1):18-31, 1980. Google Scholar
  67. Thilo Mie. Short pcpps verifiable in polylogarithmic time with o (1) queries. Annals of Mathematics and Artificial Intelligence, 56(3-4):313-338, 2009. Google Scholar
  68. Gonzalo Navarro. A guided tour to approximate string matching. ACM computing surveys (CSUR), 33(1):31-88, 2001. Google Scholar
  69. Rafail Ostrovsky and Yuval Rabani. Low distortion embeddings for edit distance. Journal of the ACM (JACM), 54(5):23, 2007. Google Scholar
  70. Mihai Patrascu. Towards polynomial lower bounds for dynamic problems. In Proceedings of the forty-second ACM symposium on Theory of computing, pages 603-610. ACM, 2010. Google Scholar
  71. Liam Roditty and Virginia Vassilevska Williams. Fast approximation algorithms for the diameter and radius of sparse graphs. In Proc. of 45th STOC, pages 515-524, 2013. Google Scholar
  72. Balaram Saha. The dyck language edit distance problem in near-linear time. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on, pages 611-620. IEEE, 2014. Google Scholar
  73. Barna Saha. Language edit distance and maximum likelihood parsing of stochastic grammars: Faster algorithms and connection to fundamental graph problems. In Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium on, pages 118-135. IEEE, 2015. Google Scholar
  74. Rajesh Santhanam and Ross Williams. On medium-uniformity and circuit lower bounds. In Computational Complexity (CCC), 2013 IEEE Conference on, pages 15-23. IEEE, 2013. Google Scholar
  75. Temple F Smith and Michael S Waterman. Identification of common molecular subsequences. Journal of molecular biology, 147(1):195-197, 1981. Google Scholar
  76. Yinglei Song. Time and space efficient algorithms for rna folding with the four-russians technique. arXiv preprint arXiv:1503.05670, 2015. Google Scholar
  77. Julie D Thompson, Desmond G Higgins, and Toby J Gibson. Clustal w: improving the sensitivity of progressive multiple sequence alignment through sequence weighting, position-specific gap penalties and weight matrix choice. Nucleic acids research, 22(22):4673-4680, 1994. Google Scholar
  78. Leslie G Valiant. Graph-theoretic arguments in low-level complexity. Springer, 1977. Google Scholar
  79. Virginia Vassilevska Williams and Ryan Williams. Subcubic equivalences between path, matrix and triangle problems. In Proc. of 51st FOCS, pages 645-654, 2010. Google Scholar
  80. Balaji Venkatachalam, Dan Gusfield, and Yelena Frid. Faster algorithms for rna-folding using the four-russians method. Algorithms for Molecular Biology, 9(1):1, 2014. Google Scholar
  81. Emanuele Viola. On the power of small-depth computation. Now Publishers Inc, 2009. Google Scholar
  82. Ryan Williams. A new algorithm for optimal constraint satisfaction and its implications. In Automata, Languages and Programming, pages 1227-1237. Springer, 2004. Google Scholar
  83. Ryan Williams. Improving exhaustive search implies superpolynomial lower bounds. SIAM Journal on Computing, 42(3):1218-1244, 2013. Google Scholar
  84. Ryan Williams. Natural proofs versus derandomization. In Proceedings of the forty-fifth annual ACM symposium on Theory of computing, pages 21-30. ACM, 2013. Google Scholar
  85. Ryan Williams. Algorithms for Circuits and Circuits for Algorithms: Connecting the Tractable and Intractable. In Proceedings of the International Congress of Mathematicians, 2014. URL: http://web.stanford.edu/~rrwill/ICM-survey.pdf.
  86. Ryan Williams. Nonuniform ACC circuit lower bounds. J. ACM, 61(1):2:1-2:32, 2014. Google Scholar
  87. Ryan Williams. Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation. In CCC'16, 2016. Google Scholar
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