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Polylogarithmic Approximation Algorithms for Weighted-F-Deletion Problems

Authors Akanksha Agrawal, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, Meirav Zehavi

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Author Details

Akanksha Agrawal
  • Institute for Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI), Budapest, Hungary
Daniel Lokshtanov
  • University of Bergen, Norway
Pranabendu Misra
  • University of Bergen, Norway
Saket Saurabh
  • Institute of Mathematical Sciences, HBNI, Chennai, India, University of Bergen, Norway, and UMI ReLax
Meirav Zehavi
  • Ben-Gurion University, Beersheba, Israel

Funding

This research has received funding from the European Research Council under ERC grant no. 306992 PARAPPROX, ERC grant no. 715744 PaPaALG, and ERC grant no. 725978 SYSTEMATICGRAPH.

Cite AsGet BibTex

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). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 1:1-1:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.1

Abstract

Let F be a family of graphs. A canonical vertex deletion problem corresponding to F is defined as follows: given an n-vertex undirected graph G and a weight function w: V(G) - >R^+, find a minimum weight subset S subseteq V(G) such that G-S belongs to F. This is known as Weighted F Vertex Deletion problem. In this paper we devise a recursive scheme to obtain O(log^{O(1)} n)-approximation algorithms for such problems, building upon the classical technique of finding balanced separators in a graph. Roughly speaking, our scheme applies to those problems, where an optimum solution S together with a well-structured set X, form a balanced separator of the input graph. In this paper, we obtain the first O(log^{O(1)} n)-approximation algorithms for the following vertex deletion problems. - Let {F} be a finite set of graphs containing a planar graph, and F=G(F) be the family of graphs such that every graph H in G(F) excludes all graphs in F as minors. The vertex deletion problem corresponding to F=G(F) is the Weighted Planar F-Minor-Free Deletion (WPF-MFD) problem. We give randomized and deterministic approximation algorithms for WPF-MFD with ratios O(log^{1.5} n) and O(log^2 n), respectively. Previously, only a randomized constant factor approximation algorithm for the unweighted version of the problem was known [FOCS 2012]. - We give an O(log^2 n)-factor approximation algorithm for Weighted Chordal Vertex Deletion (WCVD), the vertex deletion problem to the family of chordal graphs. On the way to this algorithm, we also obtain a constant factor approximation algorithm for Multicut on chordal graphs. - We give an O(log^3 n)-factor approximation algorithm for Weighted Distance Hereditary Vertex Deletion (WDHVD), also known as Weighted Rankwidth-1 Vertex Deletion (WR-1VD). This is the vertex deletion problem to the family of distance hereditary graphs, or equivalently, the family of graphs of rankwidth one. We believe that our recursive scheme can be applied to obtain O(log^{O(1)} n)-approximation algorithms for many other problems as well.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Approximation algorithms
Keywords
  • Approximation Algorithms
  • Planar- F-Deletion
  • Separator

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References

  1. Akanksha Agrawal, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi. Feedback vertex set inspired kernel for chordal vertex deletion. In Proceedings of the 28th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1383-1398, 2017. Google Scholar
  2. Akanksha Agrawal, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi. Polylogarithmic approximation algorithms for weighted-dollarbackslashmathcalFdollar-deletion problems. arXiv preprint arXiv:1707.04908, 2017. Google Scholar
  3. Vineet Bafna, Piotr Berman, and Toshihiro Fujito. A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM Journal on Discrete Mathematics, 12(3):289-297, 1999. 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 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1964-1979, 2017. Google Scholar
  5. Reuven Bar-Yehuda and Shimon Even. A linear-time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms, 2(2):198-203, 1981. Google Scholar
  6. Reuven Bar-Yehuda, Dan Geiger, Joseph Naor, and Ron M. Roth. Approximation algorithms for the feedback vertex set problem with applications to constraint satisfaction and Bayesian inference. SIAM Journal on Computing, 27(4):942-959, 1998. Google Scholar
  7. Richard B. Borie, R. Gary Parker, and Craig A. Tovey. Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families. Algorithmica, 7(5&6):555-581, 1992. Google Scholar
  8. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems, 33(2):125-150, 2000. Google Scholar
  9. Bruno Courcelle and Stephan Olariu. Upper bounds to the clique width of graphs. Discrete Applied Mathematics, 101(1-3):77-114, 2000. Google Scholar
  10. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  11. M Farber. On diameters and radii of bridged graphs. Discrete Mathematics, 73:249-260, 1989. Google Scholar
  12. 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
  13. Samuel Fiorini, Gwenaël Joret, and Ugo Pietropaoli. Hitting diamonds and growing cacti. In Proceedings of the 14th Conference on Integer Programming and Combinatorial Optimization (IPCO), volume 6080, pages 191-204, 2010. Google Scholar
  14. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, Geevarghese Philip, and Saket Saurabh. Hitting forbidden minors: Approximation and kernelization. SIAM Journal on Discrete Mathematics, 30(1):383-410, 2016. Google Scholar
  15. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Planar f-deletion: Approximation, kernelization and optimal fpt algorithms. In Proceedings of IEEE 53rd Annual Symposium on Foundations of Computer Science (FOCS), pages 470-479, 2012. Google Scholar
  16. Fedor V. Fomin, Daniel Lokshtanov, Venkatesh Raman, and Saket Saurabh. Bidimensionality and EPTAS. In Proceedings of the 22nd ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 748-759, 2011. Google Scholar
  17. Fedor V. Fomin, Daniel Lokshtanov, and Saket Saurabh. Bidimensionality and geometric graphs. In Proceedings of the 23rd ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1563-1575, 2012. Google Scholar
  18. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. Bidimensionality and kernels. In Proceedings of the 21st ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 503-510, 2010. Google Scholar
  19. Naveen Garg, Vijay V. Vazirani, and Mihalis Yannakakis. Approximate max-flow min-(multi)cut theorems and their applications. SIAM Journal on Computing, 25(2):235-251, 1996. Google Scholar
  20. Daniel Golovin, Viswanath Nagarajan, and Mohit Singh. Approximating the k-multicut problem. In ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 621-630, 2006. Google Scholar
  21. Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York, 1980. Google Scholar
  22. Anupam Gupta, Euiwoong Lee, Jason Li, Pasin Manurangsi, and Michał Włodarczyk. Losing treewidth by separating subsets. arXiv preprint arXiv:1804.01366, 2018. Google Scholar
  23. Peter L Hammer and Frédéric Maffray. Completely separable graphs. Discrete applied mathematics, 27(1):85-99, 1990. Google Scholar
  24. Petr Hlinený, Sang-il Oum, Detlef Seese, and Georg Gottlob. Width parameters beyond tree-width and their applications. The Computer Journal, 51(3):326-362, 2008. Google Scholar
  25. Edward Howorka. A characterization of distance-hereditary graphs. The quarterly journal of mathematics, 28(4):417-420, 1977. Google Scholar
  26. Bart M. P. Jansen and Marcin Pilipczuk. Approximation and kernelization for chordal vertex deletion. In Proceedings of the 28th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1399-1418, 2017. Google Scholar
  27. E. J. Kim and O. Kwon. A Polynomial Kernel for Distance-Hereditary Vertex Deletion. ArXiv e-prints, 2016. URL: http://arxiv.org/abs/1610.07229.
  28. T Leighton and S Rao. Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. Journal of the ACM, 46:787–-832, 1999. Google Scholar
  29. 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
  30. Carsten Lund and Mihalis Yannakakis. The approximation of maximum subgraph problems. In Proceedings of the 20nd International Colloquium on Automata, Languages and Programming (ICALP), volume 700, pages 40-51, 1993. Google Scholar
  31. John W Moon and Leo Moser. On cliques in graphs. Israel journal of Mathematics, 3(1):23-28, 1965. Google Scholar
  32. G. L. Nemhauser and L. E. Trotter, Jr. Properties of vertex packing and independence system polyhedra. Mathematical Programming, 6:48-61, 1974. Google Scholar
  33. Sang-il Oum. Rank-width and vertex-minors. Journal of Combinatorial Theory, Series B, 95(1):79-100, 2005. Google Scholar
  34. Sang-il Oum. Approximating rank-width and clique-width quickly. ACM Transactions on Algorithms, 5(1), 2008. Google Scholar
  35. Sang-il Oum. Rank-width: Algorithmic and structural results. CoRR, abs/1601.03800, 2016. Google Scholar
  36. Sang-il Oum and Paul D. Seymour. Approximating clique-width and branch-width. Journal of Combinatorial Theory, Series B, 96(4):514-528, 2006. Google Scholar
  37. Neil Robertson and P D Seymour. Graph minors. v. excluding a planar graph. Journal of Combinatorial Theory Series B, 41(1):92-114, 1986. Google Scholar
  38. Neil Robertson and Paul D. Seymour. Graph minors .xiii. the disjoint paths problem. Journal of Combinatorial Theory, Series B, 63(1):65-110, 1995. Google Scholar
  39. Neil Robertson and Paul D. Seymour. Graph minors. XX. wagner’s conjecture. Journal of Combinatorial Theory, Series B, 92(2):325-357, 2004. Google Scholar
  40. Horst Sachs. On the berge conjecture concerning perfect graphs. Combinatorial Structures and their Applications, 37:384, 1970. Google Scholar
  41. Shuji Tsukiyama, Mikio Ide, Hiromu Ariyoshi, and Isao Shirakawa. A new algorithm for generating all the maximal independent sets. SIAM Journal on Computing, 6(3):505-517, 1977. Google Scholar
  42. Mihalis Yannakakis. The effect of a connectivity requirement on the complexity of maximum subgraph problems. Journal of the ACM, 26(4):618-630, 1979. Google Scholar
  43. Mihalis Yannakakis. Some open problems in approximation. In Proceedings of 2nd Italian Conference on Algorithms and Complexity, Second (CIAC), pages 33-39, 1994. Google Scholar
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