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Balanced Judicious Bipartition is Fixed-Parameter Tractable

Authors Daniel Lokshtanov, Saket Saurabh, Roohani Sharma, Meirav Zehavi

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Daniel Lokshtanov
Saket Saurabh
Roohani Sharma
Meirav Zehavi

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Daniel Lokshtanov, Saket Saurabh, Roohani Sharma, and Meirav Zehavi. Balanced Judicious Bipartition is Fixed-Parameter Tractable. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 93, pp. 40:1-40:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.FSTTCS.2017.40

Abstract

The family of judicious partitioning problems, introduced by Bollob\'as and Scott to the field of extremal combinatorics, has been extensively studied from a structural point of view for over two decades. This rich realm of problems aims to counterbalance the objectives of classical partitioning problems such as Min Cut, Min Bisection and Max Cut. While these classical problems focus solely on the minimization/maximization of the number of edges crossing the cut, judicious (bi)partitioning problems ask the natural question of the minimization/maximization of the number of edges lying in the (two) sides of the cut. In particular, Judicious Bipartition (JB) seeks a bipartition that is "judicious" in the sense that neither side is burdened by too many edges, and Balanced JB also requires that the sizes of the sides themselves are "balanced" in the sense that neither of them is too large. Both of these problems were defined in the work by Bollob\'as and Scott, and have received notable scientific attention since then. In this paper, we shed light on the study of judicious partitioning problems from the viewpoint of algorithm design. Specifically, we prove that BJB is FPT (which also proves that JB is FPT).

Subject Classification

Keywords
  • Judicious Partition
  • Tree Decomposition
  • Parameterized Complexity
  • Odd Cycle Transversal
  • Minimum Bisection

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References

  1. Noga Alon, Béla Bollobás, Michael Krivelevich, and Benny Sudakov. Maximum cuts and judicious partitions in graphs without short cycles. Journal of Combinatorial Theory, Series B, 88(2):329-346, 2003. Google Scholar
  2. Florian Barbero, Gregory Gutin, Mark Jones, and Bin Sheng. Parameterized and approximation algorithms for the load coloring problem. Algorithmica, pages 1-19, 2015. Google Scholar
  3. Béla Bollobás and Alex D Scott. Judicious partitions of graphs. Periodica Mathematica Hungarica, 26(2):125-137, 1993. Google Scholar
  4. Béla Bollobás and Alex D Scott. Judicious partitions of hypergraphs. Journal of Combinatorial Theory, Series A, 78(1):15-31, 1997. Google Scholar
  5. Béla Bollobás and Alex D Scott. Exact bounds for judicious partitions of graphs. Combinatorica, 19(4):473-486, 1999. Google Scholar
  6. Béla Bollobás and Alex D Scott. Judicious partitions of 3-uniform hypergraphs. European Journal of Combinatorics, 21(3):289-300, 2000. Google Scholar
  7. Béla Bollobás and Alex D Scott. Problems and results on judicious partitions. Random Structures &Algorithms, 21(3-4):414-430, 2002. Google Scholar
  8. Béla Bollobás and Alex D Scott. Judicious partitions of bounded-degree graphs. Journal of Graph Theory, 46(2):131-143, 2004. Google Scholar
  9. Béla Bollobás and Alex D Scott. Max k-cut and judicious k-partitions. Discrete Mathematics, 310(15):2126-2139, 2010. Google Scholar
  10. Edouard Bonnet, Bruno Escoffier, Vangelis Th. Paschos, and Emeric Tourniaire. Multi-parameter analysis for local graph partitioning problems: Using greediness for parameterization. Algorithmica, 71(3):566-580, 2015. URL: http://dx.doi.org/10.1007/s00453-014-9920-6.
  11. Leizhen Cai. Parameterized complexity of cardinality constrained optimization problems. Comput. J., 51(1):102-121, 2008. URL: http://dx.doi.org/10.1093/comjnl/bxm086.
  12. Jianer Chen, Yang Liu, and Songjian Lu. An improved parameterized algorithm for the minimum node multiway cut problem. Algorithmica, 55(1):1-13, 2009. Google Scholar
  13. Rajesh Chitnis, Marek Cygan, MohammadTaghi Hajiaghayi, Marcin Pilipczuk, and Michal Pilipczuk. Designing FPT algorithms for cut problems using randomized contractions. SIAM J. Comput., 45(4):1171-1229, 2016. Google Scholar
  14. Robert Crowston, Gregory Gutin, Mark Jones, and Gabriele Muciaccia. Maximum balanced subgraph problem parameterized above lower bound. Theor. Comput. Sci., 513:53-64, 2013. Google Scholar
  15. Robert Crowston, Mark Jones, and Matthias Mnich. Max-cut parameterized above the edwards-erdős bound. Algorithmica, 72(3):734-757, 2015. Google Scholar
  16. Marek Cygan, Daniel Lokshtanov, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Minimum bisection is fixed parameter tractable. In David B. Shmoys, editor, Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 323-332. ACM, 2014. URL: http://dx.doi.org/10.1145/2591796.2591852.
  17. Christopher S Edwards. Some extremal properties of bipartite subgraphs. Canad. J. Math, 25(3):475-483, 1973. Google Scholar
  18. Christopher S Edwards. An improved lower bound for the number of edges in a largest bipartite subgraph. In Proc. 2nd Czechoslovak Symposium on Graph Theory, Prague, pages 167-181, 1975. Google Scholar
  19. Michael Etscheid and Matthias Mnich. Linear kernels and linear-time algorithms for finding large cuts. In Seok-Hee Hong, editor, 27th International Symposium on Algorithms and Computation, ISAAC 2016, December 12-14, 2016, Sydney, Australia, volume 64 of LIPIcs, pages 31:1-31:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ISAAC.2016.31.
  20. Gregory Gutin and Mark Jones. Parameterized algorithms for load coloring problem. Information Processing Letters, 114(8):446-449, 2014. Google Scholar
  21. Gregory Gutin and Anders Yeo. Note on maximal bisection above tight lower bound. Information Processing Letters, 110(21):966-969, 2010. Google Scholar
  22. Gregory Gutin and Anders Yeo. Constraint satisfaction problems parameterized above or below tight bounds: A survey. In The Multivariate Algorithmic Revolution and Beyond - Essays Dedicated to Michael R. Fellows on the Occasion of His 60th Birthday, volume 7370 of Lecture Notes in Computer Science, pages 257-286. Springer, 2012. Google Scholar
  23. John Haslegrave. Judicious partitions of uniform hypergraphs. Combinatorica, 34(5):561-572, 2014. Google Scholar
  24. Ken-ichi Kawarabayashi and Mikkel Thorup. The minimum k-way cut of bounded size is fixed-parameter tractable. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 160-169, 2011. Google Scholar
  25. Choongbum Lee, Po-Shen Loh, and Benny Sudakov. Judicious partitions of directed graphs. Random Structures &Algorithms, 48(1):147-170, 2016. Google Scholar
  26. Daniel Lokshtanov, N. S. Narayanaswamy, Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. Faster parameterized algorithms using linear programming. ACM Trans. Algorithms, 11(2):15:1-15:31, 2014. URL: http://dx.doi.org/10.1145/2566616.
  27. Daniel Lokshtanov, Saket Saurabh, Roohani Sharma, and Meirav Zehavi. Balanced judicious partition is fixed-parameter tractable. arXiv preprint arXiv:1710.05491, 2017. Google Scholar
  28. Jie Ma and Xingxing Yu. On judicious bipartitions of graphs. Combinatorica, 36(5):537-556, 2016. Google Scholar
  29. Meena Mahajan and Venkatesh Raman. Parameterizing above guaranteed values: Maxsat and maxcut. Journal of Algorithms, 31(2):335-354, 1999. Google Scholar
  30. Meena Mahajan, Venkatesh Raman, and Somnath Sikdar. Parameterizing above or below guaranteed values. J. Comput. Syst. Sci., 75(2):137-153, 2009. Google Scholar
  31. Dániel Marx. Parameterized graph separation problems. Theor. Comput. Sci., 351(3):394-406, 2006. Google Scholar
  32. Dániel Marx and Igor Razgon. Fixed-parameter tractability of multicut parameterized by the size of the cutset. SIAM J. Comput., 43(2):355-388, 2014. Google Scholar
  33. Matthias Mnich and Rico Zenklusen. Bisections above tight lower bounds. In Graph-Theoretic Concepts in Computer Science - 38th International Workshop, WG 2012, Jerusalem, Israel, June 26-28, 2012, Revised Selcted Papers, volume 7551 of Lecture Notes in Computer Science, pages 184-193. Springer, 2012. Google Scholar
  34. Saket Saurabh and Meirav Zehavi. (k, n-k)-max-cut: An 𝒪^*(2^p)-time algorithm and a polynomial kernel. In Evangelos Kranakis, Gonzalo Navarro, and Edgar Chávez, editors, LATIN 2016: Theoretical Informatics - 12th Latin American Symposium, Ensenada, Mexico, April 11-15, 2016, Proceedings, volume 9644 of Lecture Notes in Computer Science, pages 686-699. Springer, 2016. URL: http://dx.doi.org/10.1007/978-3-662-49529-2_51.
  35. Alex D Scott. Judicious partitions and related problems. Surveys in combinatorics, 327:95-117, 2005. Google Scholar
  36. Hadas Shachnai and Meirav Zehavi. Representative families: A unified tradeoff-based approach. J. Comput. Syst. Sci., 82(3):488-502, 2016. URL: http://dx.doi.org/10.1016/j.jcss.2015.11.008.
  37. Baogang Xu, Juan Yan, and Xingxing Yu. Balanced judicious bipartitions of graphs. Journal of Graph Theory, 63(3):210-225, 2010. Google Scholar
  38. Baogang Xu and Xingxing Yu. Judicious k-partitions of graphs. Journal of Combinatorial Theory, Series B, 99(2):324-337, 2009. Google Scholar
  39. Baogang Xu and Xingxing Yu. On judicious bisections of graphs. J. Comb. Theory, Ser. B, 106:30-69, 2014. Google Scholar
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