Global and Fixed-Terminal Cuts in Digraphs

Authors Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király, Euiwoong Lee, Chao Xu



PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2017.2.pdf
  • Filesize: 0.86 MB
  • 20 pages

Document Identifiers

Author Details

Kristóf Bérczi
Karthekeyan Chandrasekaran
Tamás Király
Euiwoong Lee
Chao Xu

Cite AsGet BibTex

Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király, Euiwoong Lee, and Chao Xu. Global and Fixed-Terminal Cuts in Digraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 2:1-2:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.2

Abstract

The computational complexity of multicut-like problems may vary significantly depending on whether the terminals are fixed or not. In this work we present a comprehensive study of this phenomenon in two types of cut problems in directed graphs: double cut and bicut. 1. Fixed-terminal edge-weighted double cut is known to be solvable efficiently. We show that fixed-terminal node-weighted double cut cannot be approximated to a factor smaller than 2 under the Unique Games Conjecture (UGC), and we also give a 2-approximation algorithm. For the global version of the problem, we prove an inapproximability bound of 3/2 under UGC. 2. Fixed-terminal edge-weighted bicut is known to have an approximability factor of 2 that is tight under UGC. We show that the global edge-weighted bicut is approximable to a factor strictly better than 2, and that the global node-weighted bicut cannot be approximated to a factor smaller than 3/2 under UGC. 3. In relation to these investigations, we also prove two results on undirected graphs which are of independent interest. First, we show NP-completeness and a tight inapproximability bound of 4/3 for the node-weighted 3-cut problem under UGC. Second, we show that for constant k, there exists an efficient algorithm to solve the minimum {s,t}-separating k-cut problem. Our techniques for the algorithms are combinatorial, based on LPs and based on the enumeration of approximate min-cuts. Our hardness results are based on combinatorial reductions and integrality gap instances.
Keywords
  • Directed Graphs
  • Arborescence
  • Graph Cuts
  • Hardness of Approximation

Metrics

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

References

  1. H. Angelidakis, Y. Makarychev, and P. Manurangsi. An Improved Integrality Gap for the Călinescu-Karloff-Rabani Relaxation for Multiway Cut. Preprint arXiv:1611.05530, 2016. URL: https://arxiv.org/abs/1611.05530.
  2. K. Bérczi, K. Chandrasekaran, T. Király, E. Lee, and C. Xu. Global and fixed-terminal cuts in digraphs. Preprint arXiv:1612.00156, 2017. URL: https://arxiv.org/abs/1612.00156.
  3. A. Bernáth and G. Pap. Blocking optimal arborescences. In Proceedings of the 16th International Conference on Integer Programming and Combinatorial Optimization (IPCO), pages 74-85, 2013. Google Scholar
  4. C. Chekuri and V. Madan. Simple and fast rounding algorithms for directed and node-weighted multiway cut. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'16, pages 797-807, 2016. Google Scholar
  5. C. Chekuri and V. Madan. Approximating multicut and the demand graph. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'17, 2017. Google Scholar
  6. J. Cheriyan and R. Thurimella. Fast algorithms for k-shredders and k-node connectivity augmentation. Journal of Algorithms, 33(1):15-50, 1999. Google Scholar
  7. K. Cheung, W. Cunningham, and L. Tang. Optimal 3-terminal cuts and linear programming. Mathematical Programming, 106(1):1-23, 2006. Google Scholar
  8. M. Chlebík and J. Chlebíková. Complexity of approximating bounded variants of optimization problems. Theoretical Computer Science, 354(3):320-338, 2006. Google Scholar
  9. G. Călinescu, H. Karloff, and Y. Rabani. An improved approximation algorithm for multiway cut. Journal of Computer and System Sciences, 60(3):564-574, 2000. Google Scholar
  10. E. Dahlhaus, D. Johnson, C. Papadimitriou, P. Seymour, and M. Yannakakis. The complexity of multiterminal cuts. SIAM Journal on Computing, 23(4):864-894, 1994. Google Scholar
  11. R. Erbacher, T. Jaeger, N. Talele, and J. Teutsch. Directed multicut with linearly ordered terminals. Preprint arXiv:1407.7498, 2014. URL: https://arxiv.org/abs/1407.7498.
  12. T. Fukunaga. Computing minimum multiway cuts in hypergraphs. Discrete Optimization, 10(4):371-382, 2013. Google Scholar
  13. N. Garg, V. Vazirani, and M. Yannakakis. Multiway cuts in node weighted graphs. Journal of Algorithms, 50(1):49-61, 2004. Google Scholar
  14. O. Goldschmidt and D. Hochbaum. A polynomial algorithm for the k-cut problem for fixed k. Math. Oper. Res., 19(1):24-37, Feb 1994. Google Scholar
  15. T. Jordán. On the number of shredders. Journal of Graph Theory, 31(3):195-200, 1999. Google Scholar
  16. D. Karger, P. Klein, C. Stein, M. Thorup, and N. Young. Rounding algorithms for a geometric embedding of minimum multiway cut. Mathematics of Operations Research, 29(3):436-461, 2004. Google Scholar
  17. D. Karger and R. Motwani. Derandomization through approximation. In Proceedings of the 26th annual ACM symposium on Theory of computing, STOC'94, pages 497-506, 1994. Google Scholar
  18. D. Karger and C. Stein. A new approach to the minimum cut problem. Journal of ACM, 43(4):601-640, July 1996. Google Scholar
  19. E. Lee. Improved Hardness for Cut, Interdiction, and Firefighter Problems. Preprint arXiv:1607.05133, 2016. URL: https://arxiv.org/abs/1607.05133.
  20. G. Liberman and Z. Nutov. On shredders and vertex connectivity augmentation. Journal of Discrete Algorithms, 5(1):91-101, 2007. Google Scholar
  21. R. Manokaran, J. Naor, P. Raghavendra, and R. Schwartz. SDP Gaps and UGC Hardness for Multiway Cut, 0-extension, and Metric Labeling. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing, STOC'08, pages 11-20, 2008. Google Scholar
  22. E. Mossel. Gaussian bounds for noise correlation of functions. Geometric and Functional Analysis, 19(6):1713-1756, 2010. Google Scholar
  23. J. Naor and L. Zosin. A 2-approximation algorithm for the directed multiway cut problem. SIAM Journal on Computing, 31(2):477-482, 2001. Google Scholar
  24. K. Okumoto, T. Fukunaga, and H. Nagamochi. Divide-and-conquer algorithms for partitioning hypergraphs and submodular systems. Algorithmica, 62(3):787-806, 2012. Google Scholar
  25. M. Queyranne. On Optimum k-way Partitions with Submodular Costs and Minimum Part-Size Constraints. Talk Slides, 2012. URL: https://smartech.gatech.edu/bitstream/handle/1853/43309/Queyranne.pdf.
  26. A. Schrijver. Combinatorial Optimization: Polyhedra and Efficiency. Algorithms and Combinatorics. Springer, 2003. Google Scholar
  27. A. Sharma and J. Vondrák. Multiway cut, pairwise realizable distributions, and descending thresholds. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, STOC'14, pages 724-733, 2014. Google Scholar
  28. L. Tseng and N. Vaidya. Fault-Tolerant Consensus in Directed Graphs. In Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing (PODC 2015), pages 451-460, 2015. Google Scholar
  29. L. Végh. Augmenting undirected node-connectivity by one. SIAM J. Discrete Math., 25(2):695-718, 2011. Google Scholar
  30. M. Xiao. Finding minimum 3-way cuts in hypergraphs. Information Processing Letters, 110(14):554-558, 2010. Google Scholar
  31. L. Zhao, H. Nagamochi, and T. Ibaraki. Greedy splitting algorithms for approximating multiway partition problems. Mathematical Programming, 102(1):167-183, 2005. 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