Mildly Exponential Time Approximation Algorithms for Vertex Cover, Balanced Separator and Uniform Sparsest Cut

Authors Pasin Manurangsi, Luca Trevisan



PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2018.20.pdf
  • Filesize: 0.54 MB
  • 17 pages

Document Identifiers

Author Details

Pasin Manurangsi
  • University of California, Berkeley, USA
Luca Trevisan
  • University of California, Berkeley, USA

Cite AsGet BibTex

Pasin Manurangsi and Luca Trevisan. Mildly Exponential Time Approximation Algorithms for Vertex Cover, Balanced Separator and Uniform Sparsest Cut. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.20

Abstract

In this work, we study the trade-off between the running time of approximation algorithms and their approximation guarantees. By leveraging a structure of the "hard" instances of the Arora-Rao-Vazirani lemma [Sanjeev Arora et al., 2009; James R. Lee, 2005], we show that the Sum-of-Squares hierarchy can be adapted to provide "fast", but still exponential time, approximation algorithms for several problems in the regime where they are believed to be NP-hard. Specifically, our framework yields the following algorithms; here n denote the number of vertices of the graph and r can be any positive real number greater than 1 (possibly depending on n). - A (2 - 1/(O(r)))-approximation algorithm for Vertex Cover that runs in exp (n/(2^{r^2)})n^{O(1)} time. - An O(r)-approximation algorithms for Uniform Sparsest Cut and Balanced Separator that runs in exp (n/(2^{r^2)})n^{O(1)} time. Our algorithm for Vertex Cover improves upon Bansal et al.'s algorithm [Nikhil Bansal et al., 2017] which achieves (2 - 1/(O(r)))-approximation in time exp (n/(r^r))n^{O(1)}. For Uniform Sparsest Cut and Balanced Separator, our algorithms improve upon O(r)-approximation exp (n/(2^r))n^{O(1)}-time algorithms that follow from a work of Charikar et al. [Moses Charikar et al., 2010].

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Approximation algorithms
  • Exponential-time algorithms
  • Vertex Cover
  • Sparsest Cut
  • Balanced Separator

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Amit Agarwal, Moses Charikar, Konstantin Makarychev, and Yury Makarychev. O(√log n) approximation algorithms for min UnCut, min 2CNF deletion, and directed cut problems. In STOC, pages 573-581, 2005. URL: http://dx.doi.org/10.1145/1060590.1060675.
  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: http://dx.doi.org/10.1145/2775105.
  3. Sanjeev Arora, Russell Impagliazzo, William Matthews, and David Steurer. Improved algorithms for unique games via divide and conquer. ECCC, 17:41, 2010. Google Scholar
  4. Sanjeev Arora, James R. Lee, and Assaf Naor. Euclidean distortion and the sparsest cut. In STOC, pages 553-562, 2005. URL: http://dx.doi.org/10.1145/1060590.1060673.
  5. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501-555, 1998. URL: http://dx.doi.org/10.1145/278298.278306.
  6. Sanjeev Arora, Satish Rao, and Umesh V. Vazirani. Expander flows, geometric embeddings and graph partitioning. J. ACM, 56(2):5:1-5:37, 2009. URL: http://dx.doi.org/10.1145/1502793.1502794.
  7. Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization of NP. J. ACM, 45(1):70-122, 1998. URL: http://dx.doi.org/10.1145/273865.273901.
  8. Per Austrin, Siavosh Benabbas, and Konstantinos Georgiou. Better balance by being biased: A 0.8776-approximation for max bisection. In SODA, pages 277-294, 2013. URL: http://dx.doi.org/10.1137/1.9781611973105.21.
  9. Nikhil Bansal, Parinya Chalermsook, Bundit Laekhanukit, Danupon Nanongkai, and Jesper Nederlof. New tools and connections for exponential-time approximation. CoRR, abs/1708.03515, 2017. URL: http://arxiv.org/abs/1708.03515.
  10. Nikhil Bansal and Subhash Khot. Optimal long code test with one free bit. In FOCS, pages 453-462, 2009. URL: http://dx.doi.org/10.1109/FOCS.2009.23.
  11. R. Bar-Yehuda and S. Even. A local-ratio theorem for approximating the weighted vertex cover problem. In G. Ausiello and M. Lucertini, editors, Analysis and Design of Algorithms for Combinatorial Problems, volume 109 of North-Holland Mathematics Studies, pages 27-45. North-Holland, 1985. URL: http://dx.doi.org/10.1016/S0304-0208(08)73101-3.
  12. Boaz Barak, Fernando G. S. L. Brandão, Aram Wettroth Harrow, Jonathan A. Kelner, David Steurer, and Yuan Zhou. Hypercontractivity, sum-of-squares proofs, and their applications. In STOC, pages 307-326, 2012. URL: http://dx.doi.org/10.1145/2213977.2214006.
  13. Boaz Barak, Prasad Raghavendra, and David Steurer. Rounding semidefinite programming hierarchies via global correlation. In FOCS, pages 472-481, 2011. URL: http://dx.doi.org/10.1109/FOCS.2011.95.
  14. Boaz Barak and David Steurer. Sum-of-squares proofs and the quest toward optimal algorithms. ECCC, 21:59, 2014. URL: http://eccc.hpi-web.de/report/2014/059.
  15. 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.
  16. Moses Charikar, Mohammad Taghi Hajiaghayi, Howard J. Karloff, and Satish Rao. 𝓁₂² spreading metrics for vertex ordering problems. Algorithmica, 56(4):577-604, 2010. URL: http://dx.doi.org/10.1007/s00453-008-9191-1.
  17. Moses Charikar, Konstantin Makarychev, and Yury Makarychev. Local global tradeoffs in metric embeddings. SIAM J. Comput., 39(6):2487-2512, 2010. URL: http://dx.doi.org/10.1137/070712080.
  18. Shuchi Chawla, Robert Krauthgamer, Ravi Kumar, Yuval Rabani, and D. Sivakumar. On the hardness of approximating multicut and sparsest-cut. Computational Complexity, 15(2):94-114, 2006. URL: http://dx.doi.org/10.1007/s00037-006-0210-9.
  19. Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Towards a proof of the 2-to-1 games conjecture? ECCC, 23:198, 2016. URL: http://eccc.hpi-web.de/report/2016/198.
  20. Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. On non-optimally expanding sets in grassmann graphs. ECCC, 24:94, 2017. URL: https://eccc.weizmann.ac.il/report/2017/094.
  21. Irit Dinur and Shmuel Safra. On the hardness of approximating minimum vertex cover. Annals of Mathematics, 162(1):439-485, 2005. Google Scholar
  22. Uriel Feige. Relations between average case complexity and approximation complexity. In STOC, pages 534-543, 2002. URL: http://dx.doi.org/10.1145/509907.509985.
  23. Uriel Feige, MohammadTaghi Hajiaghayi, and James R. Lee. Improved approximation algorithms for minimum weight vertex separators. SIAM J. Comput., 38(2):629-657, 2008. URL: http://dx.doi.org/10.1137/05064299X.
  24. Uriel Feige and James R. Lee. An improved approximation ratio for the minimum linear arrangement problem. Inf. Process. Lett., 101(1):26-29, 2007. URL: http://dx.doi.org/10.1016/j.ipl.2006.07.009.
  25. Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  26. Dima Grigoriev. Complexity of positivstellensatz proofs for the knapsack. Computational Complexity, 10(2):139-154, 2001. URL: http://dx.doi.org/10.1007/s00037-001-8192-0.
  27. Venkatesan Guruswami and Ali Kemal Sinop. Lasserre hierarchy, higher eigenvalues, and approximation schemes for graph partitioning and quadratic integer programming with PSD objectives. In FOCS, pages 482-491, 2011. URL: http://dx.doi.org/10.1109/FOCS.2011.36.
  28. Eran Halperin. Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs. SIAM J. Comput., 31(5):1608-1623, 2002. URL: http://dx.doi.org/10.1137/S0097539700381097.
  29. Johan Håstad. Some optimal inapproximability results. J. ACM, 48(4):798-859, 2001. URL: http://dx.doi.org/10.1145/502090.502098.
  30. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001. URL: http://dx.doi.org/10.1006/jcss.2000.1727.
  31. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: http://dx.doi.org/10.1006/jcss.2001.1774.
  32. George Karakostas. A better approximation ratio for the vertex cover problem. ACM Trans. Algorithms, 5(4):41:1-41:8, 2009. URL: http://dx.doi.org/10.1145/1597036.1597045.
  33. Subhash Khot. On the power of unique 2-prover 1-round games. In CCC, page 25, 2002. URL: http://dx.doi.org/10.1109/CCC.2002.1004334.
  34. Subhash Khot. Ruling out PTAS for graph min-bisection, dense k-subgraph, and bipartite clique. SIAM J. Comput., 36(4):1025-1071, 2006. URL: http://dx.doi.org/10.1137/S0097539705447037.
  35. Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O'Donnell. Optimal inapproximability results for MAX-CUT and other 2-variable csps? SIAM J. Comput., 37(1):319-357, 2007. URL: http://dx.doi.org/10.1137/S0097539705447372.
  36. Subhash Khot, Dor Minzer, and Muli Safra. On independent sets, 2-to-2 games, and grassmann graphs. In STOC, pages 576-589, 2017. URL: http://dx.doi.org/10.1145/3055399.3055432.
  37. Subhash Khot, Dor Minzer, and Muli Safra. Pseudorandom sets in grassmann graph have near-perfect expansion. ECCC, 25:6, 2018. URL: https://eccc.weizmann.ac.il/report/2018/006.
  38. Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2-epsilon. J. Comput. Syst. Sci., 74(3):335-349, 2008. URL: http://dx.doi.org/10.1016/j.jcss.2007.06.019.
  39. Subhash Khot and Nisheeth K. Vishnoi. The unique games conjecture, integrality gap for cut problems and embeddability of negative-type metrics into 𝓁₁. J. ACM, 62(1):8:1-8:39, 2015. URL: http://dx.doi.org/10.1145/2629614.
  40. Pravesh K. Kothari, Ryuhei Mori, Ryan O'Donnell, and David Witmer. Sum of squares lower bounds for refuting any CSP. In STOC, pages 132-145, 2017. Google Scholar
  41. Jean B. Lasserre. An explicit equivalent positive semidefinite program for nonlinear 0-1 programs. SIAM Journal on Optimization, 12(3):756-769, 2002. URL: http://dx.doi.org/10.1137/S1052623400380079.
  42. James R. Lee. On distance scales, embeddings, and efficient relaxations of the cut cone. In SODA, pages 92-101, 2005. URL: http://dl.acm.org/citation.cfm?id=1070432.1070446.
  43. Frank Thomson Leighton and Satish Rao. Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. J. ACM, 46(6):787-832, 1999. URL: http://dx.doi.org/10.1145/331524.331526.
  44. Pasin Manurangsi and Prasad Raghavendra. A birthday repetition theorem and complexity of approximating dense CSPs. CoRR, abs/1607.02986, 2016. URL: http://arxiv.org/abs/1607.02986,
  45. Burkhard Monien and Ewald Speckenmeyer. Ramsey numbers and an approximation algorithm for the vertex cover problem. Acta Inf., 22(1):115-123, 1985. URL: http://dx.doi.org/10.1007/BF00290149.
  46. Dana Moshkovitz and Ran Raz. Two-query PCP with subconstant error. J. ACM, 57(5):29:1-29:29, 2010. URL: http://dx.doi.org/10.1145/1754399.1754402.
  47. Yurii Nesterov. Squared functional systems and optimization problems. In High performance optimization, pages 405-440. Springer, 2000. Google Scholar
  48. Ryan O'Donnell and Yuan Zhou. Approximability and proof complexity. In SODA, pages 1537-1556, 2013. URL: http://dx.doi.org/10.1137/1.9781611973105.111.
  49. Pablo A. Parrilo. Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, California Institute of Technology, 2000. Google Scholar
  50. Prasad Raghavendra and David Steurer. Graph expansion and the unique games conjecture. In STOC, pages 755-764, 2010. URL: http://dx.doi.org/10.1145/1806689.1806788.
  51. Prasad Raghavendra, David Steurer, and Madhur Tulsiani. Reductions between expansion problems. In CCC, pages 64-73, 2012. URL: http://dx.doi.org/10.1109/CCC.2012.43.
  52. Prasad Raghavendra and Ning Tan. Approximating CSPs with global cardinality constraints using SDP hierarchies. In SODA, pages 373-387, 2012. URL: http://portal.acm.org/citation.cfm?id=2095149&CFID=63838676&CFTOKEN=79617016.
  53. Grant Schoenebeck. Linear level lasserre lower bounds for certain k-csps. In FOCS, pages 593-602, 2008. URL: http://dx.doi.org/10.1109/FOCS.2008.74.
  54. Yuichi Yoshida and Yuan Zhou. Approximation schemes via sherali-adams hierarchy for dense constraint satisfaction problems and assignment problems. In ITCS, pages 423-438, 2014. URL: http://dx.doi.org/10.1145/2554797.2554836.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail