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# Packing Cycles Faster Than Erdos-Posa

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## Cite As

Daniel Lokshtanov, Amer E. Mouawad, Saket Saurabh, and Meirav Zehavi. Packing Cycles Faster Than Erdos-Posa. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 71:1-71:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ICALP.2017.71

## Abstract

The Cycle Packing problem asks whether a given undirected graph G=(V,E) contains k vertex-disjoint cycles. Since the publication of the classic Erdos-Posa theorem in 1965, this problem received significant scientific attention in the fields of Graph Theory and Algorithm Design. In particular, this problem is one of the first problems studied in the framework of Parameterized Complexity. The non-uniform fixed-parameter tractability of Cycle Packing follows from the Robertson–Seymour theorem, a fact already observed by Fellows and Langston in the 1980s. In 1994, Bodlaender showed that Cycle Packing can be solved in time 2^{O(k^2)}|V| using exponential space. In case a solution exists, Bodlaender's algorithm also outputs a solution (in the same time). It has later become common knowledge that Cycle Packing admits a 2^{O(k\log^2 k)}|V|-time (deterministic) algorithm using exponential space, which is a consequence of the Erdos-Posa theorem. Nowadays, the design of this algorithm is given as an exercise in textbooks on Parameterized Complexity. Yet, no algorithm that runs in time 2^{o(k\log^2k)}|V|^{O(1)}, beating the bound 2^{O(k\log^2k)}\cdot |V|^{O(1)}, has been found. In light of this, it seems natural to ask whether the 2^{O(k\log^2k)}|V|^{O(1)}\$ bound is essentially optimal. In this paper, we answer this question negatively by developing a 2^{O(k\log^2k/log log k})} |V|-time (deterministic) algorithm for Cycle Packing. In case a solution exists, our algorithm also outputs a solution (in the same time). Moreover, apart from beating the known bound, our algorithm runs in time linear in |V|, and its space complexity is polynomial in the input size.
##### Keywords
• Parameterized Complexity
• Graph Algorithms
• Cycle Packing
• Erdös-Pósa Theorem

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## References

1. Noga Alon, Shlomo Hoory, and Nathan Linial. The Moore bound for irregular graphs. Graphs and Combinatorics, 18(1):53-57, 2002. URL: http://dx.doi.org/10.1007/s003730200002.
2. Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernel bounds for path and cycle problems. Theor. Comput. Sci., 511:117-136, 2013. URL: http://dx.doi.org/10.1016/j.tcs.2012.09.006.
3. Hans L. Bodlaender, Eelko Penninkx, and Richard B. Tan. A linear kernel for the k-disjoint cycle problem on planar graphs. In Seok-Hee Hong, Hiroshi Nagamochi, and Takuro Fukunaga, editors, Algorithms and Computation, 19th International Symposium, ISAAC 2008, Gold Coast, Australia, December 15-17, 2008. Proceedings, volume 5369 of Lecture Notes in Computer Science, pages 306-317. Springer, 2008. URL: http://dx.doi.org/10.1007/978-3-540-92182-0_29.
4. Hans L. Bodlaender, Stéphan Thomassé, and Anders Yeo. Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci., 412(35):4570-4578, 2011. URL: http://dx.doi.org/10.1016/j.tcs.2011.04.039.
5. H. L. Bodlaender. On disjoint cycles. Int. J. Found. Comput. Sci., 5(1):59-68, 1994.
6. Chandra Chekuri and Julia Chuzhoy. Large-treewidth graph decompositions and applications. In Dan Boneh, Tim Roughgarden, and Joan Feigenbaum, editors, Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 291-300. ACM, 2013. URL: http://dx.doi.org/10.1145/2488608.2488645.
7. Chandra Chekuri and Julia Chuzhoy. Polynomial bounds for the grid-minor theorem. J. ACM, 63(5):40:1-40:65, 2016.
8. Julia Chuzhoy. Excluded grid theorem: Improved and simplified. In Rocco A. Servedio and Ronitt Rubinfeld, editors, Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 645-654. ACM, 2015. URL: http://dx.doi.org/10.1145/2746539.2746551.
9. Julia Chuzhoy. Excluded grid theorem: Improved and simplified (invited talk). In Rasmus Pagh, editor, 15th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2016, June 22-24, 2016, Reykjavik, Iceland, volume 53 of LIPIcs, pages 31:1-31:1. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.SWAT.2016.31.
10. M. Cygan, F. V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized algorithms. Springer, 2015.
11. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. In Rafail Ostrovsky, editor, IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 150-159. IEEE Computer Society, 2011. URL: http://dx.doi.org/10.1109/FOCS.2011.23.
12. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012.
13. Frederic Dorn, Fedor V. Fomin, and Dimitrios M. Thilikos. Catalan structures and dynamic programming in h-minor-free graphs. J. Comput. Syst. Sci., 78(5):1606-1622, 2012. URL: http://dx.doi.org/10.1016/j.jcss.2012.02.004.
14. R. Downey and M. Fellows. Fundamentals of parameterized complexity. Springer, 2013.
15. P. Erdős and L. Pósa. On independent circuits contained in a graph. Canad. J. Math., 17:347–352, 1965.
16. Fedor V. Fomin, Daniel Lokshtanov, Venkatesh Raman, and Saket Saurabh. Bidimensionality and EPTAS. In Dana Randall, editor, Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011, pages 748-759. SIAM, 2011. URL: http://dx.doi.org/10.1137/1.9781611973082.59.
17. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. Bidimensionality and kernels. In Moses Charikar, editor, Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, 2010, pages 503-510. SIAM, 2010. URL: http://dx.doi.org/10.1137/1.9781611973075.43.
18. Zachary Friggstad and Mohammad R. Salavatipour. Approximability of packing disjoint cycles. Algorithmica, 60(2):395-400, 2011. URL: http://dx.doi.org/10.1007/s00453-009-9349-5.
19. Martin Grohe and Magdalena Grüber. Parameterized approximability of the disjoint cycle problem. In Lars Arge, Christian Cachin, Tomasz Jurdzinski, and Andrzej Tarlecki, editors, Automata, Languages and Programming, 34th International Colloquium, ICALP 2007, Wroclaw, Poland, July 9-13, 2007, Proceedings, volume 4596 of Lecture Notes in Computer Science, pages 363-374. Springer, 2007. URL: http://dx.doi.org/10.1007/978-3-540-73420-8_33.
20. R. Impagliazzo and R. Paturi. On the complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367–375, 2001.
21. Alon Itai and Michael Rodeh. Finding a minimum circuit in a graph. SIAM J. Comput., 7(4):413-423, 1978. URL: http://dx.doi.org/10.1137/0207033.
22. Felix Joos. Parity linkage and the erdős-pósa property of odd cycles through prescribed vertices in highly connected graphs. In Ernst W. Mayr, editor, Graph-Theoretic Concepts in Computer Science - 41st International Workshop, WG 2015, Garching, Germany, June 17-19, 2015, Revised Papers, volume 9224 of Lecture Notes in Computer Science, pages 339-350. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-53174-7_24.
23. Naonori Kakimura and Ken-ichi Kawarabayashi. Half-integral packing of odd cycles through prescribed vertices. Combinatorica, 33(5):549-572, 2013. URL: http://dx.doi.org/10.1007/s00493-013-2865-6.
24. Naonori Kakimura, Ken-ichi Kawarabayashi, and Dániel Marx. Packing cycles through prescribed vertices. J. Comb. Theory, Ser. B, 101(5):378-381, 2011. URL: http://dx.doi.org/10.1016/j.jctb.2011.03.004.
25. Ken-ichi Kawarabayashi and Yusuke Kobayashi. Fixed-parameter tractability for the subset feedback set problem and the s-cycle packing problem. J. Comb. Theory, Ser. B, 102(4):1020-1034, 2012. URL: http://dx.doi.org/10.1016/j.jctb.2011.12.001.
26. Ken-ichi Kawarabayashi, Daniel Král', Marek Krcál, and Stephan Kreutzer. Packing directed cycles through a specified vertex set. In Sanjeev Khanna, editor, Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 365-377. SIAM, 2013. URL: http://dx.doi.org/10.1137/1.9781611973105.27.
27. Ken-ichi Kawarabayashi and Atsuhiro Nakamoto. The erdos-pósa property for vertex- and edge-disjoint odd cycles in graphs on orientable surfaces. Discrete Mathematics, 307(6):764-768, 2007. URL: http://dx.doi.org/10.1016/j.disc.2006.07.008.
28. Ken-ichi Kawarabayashi and Bruce A. Reed. Highly parity linked graphs. Combinatorica, 29(2):215-225, 2009. URL: http://dx.doi.org/10.1007/s00493-009-2178-y.
29. Ken-ichi Kawarabayashi and Paul Wollan. Non-zero disjoint cycles in highly connected group labelled graphs. J. Comb. Theory, Ser. B, 96(2):296-301, 2006. URL: http://dx.doi.org/10.1016/j.jctb.2005.08.001.
30. Michael Krivelevich, Zeev Nutov, Mohammad R. Salavatipour, Jacques Yuster, and Raphael Yuster. Approximation algorithms and hardness results for cycle packing problems. ACM Trans. Algorithms, 3(4):48, 2007. URL: http://dx.doi.org/10.1145/1290672.1290685.
31. D. Lokshtanov, F. Panolan, M. S. Ramanujan, and S. Saurabh. Lossy kernelization. arXiv:1604.04111v2, to appear in STOC 2017.
32. Jesper Nederlof. Fast polynomial-space algorithms using inclusion-exclusion. Algorithmica, 65(4):868-884, 2013. URL: http://dx.doi.org/10.1007/s00453-012-9630-x.
33. M. Pontecorvi and Paul Wollan. Disjoint cycles intersecting a set of vertices. J. Comb. Theory, Ser. B, 102(5):1134-1141, 2012. URL: http://dx.doi.org/10.1016/j.jctb.2012.05.004.
34. Venkatesh Raman, Saket Saurabh, and C. R. Subramanian. Faster fixed parameter tractable algorithms for finding feedback vertex sets. ACM Trans. Algorithms, 2(3):403-415, 2006. URL: http://dx.doi.org/10.1145/1159892.1159898.
35. Dieter Rautenbach and Bruce A. Reed. The Erdős-Pósa property for odd cycles in highly connected graphs. Combinatorica, 21(2):267-278, 2001. URL: http://dx.doi.org/10.1007/s004930100024.
36. Bruce A. Reed. Mangoes and blueberries. Combinatorica, 19(2):267-296, 1999. URL: http://dx.doi.org/10.1007/s004930050056.
37. Neil Robertson and Paul D. Seymour. Graph minors. V. Excluding a planar graph. J. Comb. Theory, Ser. B, 41(1):92-114, 1986. URL: http://dx.doi.org/10.1016/0095-8956(86)90030-4.
38. Neil Robertson and Paul D. Seymour. Graph minors .xiii. the disjoint paths problem. J. Comb. Theory, Ser. B, 63(1):65-110, 1995. URL: http://dx.doi.org/10.1006/jctb.1995.1006.
39. Aleksandrs Slivkins. Parameterized tractability of edge-disjoint paths on directed acyclic graphs. SIAM J. Discrete Math., 24(1):146-157, 2010. URL: http://dx.doi.org/10.1137/070697781.
40. Carsten Thomassen. The Erdős-Pósa property for odd cycles in graphs of large connectivity. Combinatorica, 21(2):321-333, 2001. URL: http://dx.doi.org/10.1007/s004930100028.
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