Document Open Access Logo

Structural Rounding: Approximation Algorithms for Graphs Near an Algorithmically Tractable Class

Authors Erik D. Demaine, Timothy D. Goodrich, Kyle Kloster, Brian Lavallee, Quanquan C. Liu, Blair D. Sullivan, Ali Vakilian, Andrew van der Poel



PDF
Thumbnail PDF

File

LIPIcs.ESA.2019.37.pdf
  • Filesize: 0.57 MB
  • 15 pages

Document Identifiers

Author Details

Erik D. Demaine
  • MIT, Cambridge, MA, USA
Timothy D. Goodrich
  • NC State University, Raleigh, NC, USA
Kyle Kloster
  • NC State University, Raleigh, NC, USA
Brian Lavallee
  • NC State University, Raleigh, NC, USA
Quanquan C. Liu
  • MIT, Cambridge, MA, USA
Blair D. Sullivan
  • NC State University, Raleigh, NC, USA
Ali Vakilian
  • MIT, Cambridge, MA, USA
Andrew van der Poel
  • NC State University, Raleigh, NC, USA

Acknowledgements

We thank Abida Haque and Adam Hesterberg for helpful initial discussions, Nicole Wein for providing helpful references on bounded degeneracy problems, and Michael O'Brien for helpful comments on the manuscript.

Cite AsGet BibTex

Erik D. Demaine, Timothy D. Goodrich, Kyle Kloster, Brian Lavallee, Quanquan C. Liu, Blair D. Sullivan, Ali Vakilian, and Andrew van der Poel. Structural Rounding: Approximation Algorithms for Graphs Near an Algorithmically Tractable Class. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 37:1-37:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ESA.2019.37

Abstract

We develop a framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when working with real-world networks) while still guaranteeing approximation ratios. The idea is to edit a given graph via vertex- or edge-deletions to put the graph into an algorithmically tractable class, apply known approximation algorithms for that class, and then lift the solution to apply to the original graph. We give a general characterization of when an optimization problem is amenable to this approach, and show that it includes many well-studied graph problems, such as Independent Set, Vertex Cover, Feedback Vertex Set, Minimum Maximal Matching, Chromatic Number, (l-)Dominating Set, Edge (l-)Dominating Set, and Connected Dominating Set. To enable this framework, we develop new editing algorithms that find the approximately-fewest edits required to bring a given graph into one of a few important graph classes (in some cases these are bicriteria algorithms which simultaneously approximate both the number of editing operations and the target parameter of the family). For bounded degeneracy, we obtain an O(r log{n})-approximation and a bicriteria (4,4)-approximation which also extends to a smoother bicriteria trade-off. For bounded treewidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w}))-approximation, and for bounded pathwidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w} * log n))-approximation. For treedepth 2 (related to bounded expansion), we obtain a 4-approximation. We also prove complementary hardness-of-approximation results assuming P != NP: in particular, these problems are all log-factor inapproximable, except the last which is not approximable below some constant factor 2 (assuming UGC).

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • structural rounding
  • graph editing
  • approximation algorithms

Metrics

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

References

  1. Akanksha Agrawal, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi. Polylogarithmic Approximation Algorithms for Weighted-F-Deletion Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2018, August 20-22, 2018 - Princeton, NJ, USA, pages 1:1-1:15, 2018. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.1.
  2. Jochen Alber and Rolf Niedermeier. Improved Tree Decomposition Based Algorithms for Domination-like Problems. In Proceedings of the 5th Latin American Symposium on Theoretical Informatics, pages 613-627. Springer, 2001. Google Scholar
  3. Eyal Amir. Approximation Algorithms for Treewidth. Algorithmica, 56(4):448-479, 2010. Google Scholar
  4. Nikhil Bansal, Daniel Reichman, and Seeun William Umboh. LP-based robust algorithms for noisy minor-free and bounded treewidth graphs. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1964-1979. Society for Industrial and Applied Mathematics, 2017. Google Scholar
  5. Nikhil Bansal and Seeun William Umboh. Tight approximation bounds for dominating set on graphs of bounded arboricity. Information Processing Letters, 122:21-24, 2017. Google Scholar
  6. Reuven Bar-Yehuda, Keren Bendel, Ari Freund, and Dror Rawitz. Local ratio: A unified framework for approximation algorithms. ACM Computing Surveys, 36(4):422-463, 2004. Google Scholar
  7. Therese Biedl, Markus Chimani, Martin Derka, and Petra Mutzel. Crossing number for graphs with bounded pathwidth. In 28th International Symposium on Algorithms and Computation, (ISAAC), pages 13:1-13:13, Phuket, Thailand, December 2017. Google Scholar
  8. Hans L. Bodlaender, John R. Gilbert, Hjálmtyr Hafsteinsson, and Ton Kloks. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. Journal of Algorithms, 18(2):238-255, 1995. Google Scholar
  9. Geoff Boeing. Planarity and Street Network Representation in Urban Form Analysis. arXiv preprint arXiv:1806.01805, 2018. URL: http://arxiv.org/abs/1806.01805.
  10. Glencora Borradaile and Hung Le. Optimal Dynamic Program for r-Domination Problems over Tree Decompositions. In Proceedings of the 11th International Symposium on Parameterized and Exact Computation, volume 63 of Leibniz International Proceedings in Informatics (LIPIcs), pages 8:1-8:23, 2017. Google Scholar
  11. Leizhen Cai. Parameterized complexity of vertex colouring. Discrete Applied Mathematics, 127(3):415-429, 2003. Google Scholar
  12. Timothy M Chan and Sariel Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discrete & Computational Geometry, 48(2):373-392, 2012. Google Scholar
  13. Chandra Chekuri and Anastasios Sidiropoulos. Approximation algorithms for Euler genus and related problems. In Proceedings of the IEEE 54th Annual Symposium on Foundations of Computer Science, pages 167-176. IEEE, 2013. Google Scholar
  14. Julia Chuzhoy. An algorithm for the graph crossing number problem. In Proceedings of the 43rd Annual ACM Symposium on Theory of Computing, pages 303-312, 2011. Google Scholar
  15. Julia Chuzhoy, Yury Makarychev, and Anastasios Sidiropoulos. On graph crossing number and edge planarization. In Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1050-1069, 2011. Google Scholar
  16. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, Joham MM van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. In Foundations of computer science (focs), 2011 ieee 52nd annual symposium on, pages 150-159. IEEE, 2011. Google Scholar
  17. Konrad K Dabrowski, Petr A Golovach, Pim van't Hof, Daniël Paulusma, and Dimitrios M Thilikos. Editing to a planar graph of given degrees. In Computer Science-Theory and Applications, pages 143-156. Springer, 2015. Google Scholar
  18. Erik D. Demaine, Timothy D. Goodrich, Kyle Kloster, Brian Lavallee, Quanquan C. Liu, Blair D. Sullivan, Ali Vakilian, and Andrew van der Poel. Structural Rounding: Approximation Algorithms for Graphs Near an Algorithmically Tractable Class. CoRR, abs/1806.02771, 2018. URL: http://arxiv.org/abs/1806.02771.
  19. Rodney G Downey and Michael R Fellows. Fundamentals of parameterized complexity, volume 4. Springer, 2013. Google Scholar
  20. Pål Grønås Drange. Parameterized Graph Modification Algorithms. PhD thesis, The University of Bergen, 2015. Google Scholar
  21. Pål Grønås Drange, Markus S. Dregi, and R. B. Sandeep. Compressing Bounded Degree Graphs. In LATIN, volume 9644 of Lecture Notes in Computer Science, pages 362-375. Springer, 2016. Google Scholar
  22. Pål Grønås Drange, Markus Sortland Dregi, Daniel Lokshtanov, and Blair D. Sullivan. On the Threshold of Intractability. In Proceedings of the 23rd Annual European Symposium on Algorithms, pages 411-423, Patras, Greece, September 2015. Google Scholar
  23. Tomáš Ebenlendr, Petr Kolman, and Jirı Sgall. An Approximation Algorithm for Bounded Degree Deletion. Preprint, unpublished. URL: http://kam.mff.cuni.cz/~kolman/papers/star.pdf.
  24. Uriel Feige, MohammadTaghi Hajiaghayi, and James R. Lee. Improved Approximation Algorithms for Minimum Weight Vertex Separators. SIAM Journal on Computing, 38(2):629-657, 2008. Google Scholar
  25. Fedor V. Fomin, Stefan Kratsch, Marcin Pilipczuk, Michal Pilipczuk, and Yngve Villanger. Tight bounds for parameterized complexity of Cluster Editing with a small number of clusters. Journal of Computer and System Sciences, 80(7):1430-1447, 2014. Google Scholar
  26. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Planar F-deletion: Approximation, kernelization and optimal FPT algorithms. In Proceedings of the IEEE 53rd Annual Symposium on Foundations of Computer Science, pages 470-479, 2012. Google Scholar
  27. Toshihiro Fujito. A unified approximation algorithm for node-deletion problems. Discrete Applied Mathematics, 86(2-3):213-231, 1998. Google Scholar
  28. Xinbo Gao, Bing Xiao, Dacheng Tao, and Xuelong Li. A survey of graph edit distance. Pattern Analysis and Applications, 13(1):113-129, 2010. Google Scholar
  29. Michael T Gastner and Mark EJ Newman. The spatial structure of networks. The European Physical Journal B-Condensed Matter and Complex Systems, 49(2):247-252, 2006. Google Scholar
  30. Jiong Guo, Falk Hüffner, and Rolf Niedermeier. A structural view on parameterizing problems: Distance from triviality. In International Workshop on Parameterized and Exact Computation, pages 162-173. Springer, 2004. Google Scholar
  31. Anupam Gupta, Euiwoong Lee, Jason Li, Pasin Manurangsi, and Michał Włodarczyk. Losing Treewidth by Separating Subsets. CoRR, abs/1804.01366, 2018. URL: http://arxiv.org/abs/1804.01366.
  32. Venkatesan Guruswami and Euiwoong Lee. Inapproximability of H-transversal/packing. SIAM Journal on Discrete Mathematics, 31(3):1552-1571, 2017. Google Scholar
  33. Falk Hüffner, Christian Komusiewicz, and André Nichterlein. Editing Graphs Into Few Cliques: Complexity, Approximation, and Kernelization Schemes. In Proceedings of the 14th International Symposium on Algorithms and Data Structures, pages 410-421. Springer, 2015. Google Scholar
  34. Bart M. P. Jansen, Daniel Lokshtanov, and Saket Saurabh. A near-optimal planarization algorithm. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1802-1811, 2014. Google Scholar
  35. Ken-ichi Kawarabayashi. Planarity allowing few error vertices in linear time. In Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, pages 639-648. IEEE, 2009. Google Scholar
  36. Ken-ichi Kawarabayashi and Anastasios Sidiropoulos. Polylogarithmic Approximation for Minimum Planarization (Almost). In Proceedings of the 58th IEEE Annual Symposium on Foundations of Computer Science, pages 779-788, Berkeley, CA, October 2017. Google Scholar
  37. Christian Komusiewicz. Tight Running Time Lower Bounds for Vertex Deletion Problems. TOCT, 10(2):6:1-6:18, 2018. Google Scholar
  38. Michal Kotrbčík, Rastislav Královič, and Sebastian Ordyniak. Edge-Editing to a Dense and a Sparse Graph Class. In Proceedings of the 12th Latin American Symposium on Theoretical Informatics, pages 562-575. Springer, 2016. Google Scholar
  39. Mukkai S Krishnamoorthy and Narsingh Deo. Node-deletion NP-complete problems. SIAM Journal on Computing, 8(4):619-625, 1979. Google Scholar
  40. Euiwoong Lee. Partitioning a graph into small pieces with applications to path transversal. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1546-1558. Society for Industrial and Applied Mathematics, 2017. Google Scholar
  41. John M. Lewis. On the complexity of the maximum subgraph problem. In Proceedings of the 10th Annual ACM Symposium on Theory of Computing, pages 265-274. ACM, 1978. Google Scholar
  42. John M. Lewis and Mihalis Yannakakis. The node-deletion problem for hereditary properties is NP-complete. Journal of Computer and System Sciences, 20(2):219-230, 1980. Google Scholar
  43. Carsten Lund and Mihalis Yannakakis. The approximation of maximum subgraph problems. In Proceedings of the International Colloquium on Automata, Languages, and Programming, pages 40-51. Springer, 1993. Google Scholar
  44. Avner Magen and Mohammad Moharrami. Robust algorithms for on minor-free graphs based on the Sherali-Adams hierarchy. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 258-271. Springer, 2009. Google Scholar
  45. Dániel Marx. Parameterized coloring problems on chordal graphs. Theoretical Computer Science, 351(3):407-424, 2006. Google Scholar
  46. Dániel Marx. Parameterized complexity and approximation algorithms. The Computer Journal, 51(1):60-78, 2008. Google Scholar
  47. Dániel Marx and Ildikó Schlotter. Obtaining a planar graph by vertex deletion. Algorithmica, 62(3-4):807-822, 2012. Google Scholar
  48. Luke Mathieson. The parameterized complexity of editing graphs for bounded degeneracy. Theoretical Computer Science, 411(34):3181-3187, 2010. Google Scholar
  49. Luke Mathieson and Stefan Szeider. Parameterized graph editing with chosen vertex degrees. In Combinatorial Optimization and Applications, pages 13-22. Springer, 2008. Google Scholar
  50. David W. Matula and Leland L. Beck. Smallest-last ordering and clustering and graph coloring algorithms. Journal of the ACM, 30(3):417-427, 1983. Google Scholar
  51. J. Nešetřil and P. O. de Mendez. Sparsity: Graphs, Structures, and Algorithms. Algorithms and Combinatorics. Springer Berlin Heidelberg, 2012. Google Scholar
  52. Michael Okun and Amnon Barak. A new approach for approximating node deletion problems. Information Processing Letters, 88(5):231-236, 2003. Google Scholar
  53. Bruce A. Reed. Finding Approximate Separators and Computing Tree Width Quickly. In Proceedings of the 24th Annual ACM Symposium on Theory of Computing, pages 221-228, 1992. Google Scholar
  54. Marcus Schaefer. The graph crossing number and its variants: A survey. The Electronic Journal of Combinatorics, 1000:21-22, 2013. Google Scholar
  55. Mingyu Xiao. A Parameterized Algorithm for Bounded-Degree Vertex Deletion. In Proceedings of the 22nd International Conference on Computing and Combinatorics, pages 79-91, 2016. Google Scholar
  56. Mihalis Yannakakis. Node-and edge-deletion NP-complete problems. In Proceedings of the 10th Annual ACM Symposium on Theory of Computing, pages 253-264, 1978. Google Scholar
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