Document Open Access Logo

Approximating Requirement Cut via a Configuration LP

Authors Roy Schwartz, Yotam Sharoni

PDF

File

Thumbnail PDF
PDF
LIPIcs.APPROX-RANDOM.2020.53.pdf
  • Filesize: 0.56 MB
  • 16 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Facility location and clustering
  • Mathematics of computing → Combinatorial optimization
Keywords
  • Approximation
  • Requirement Cut
  • Sparsest Cut
  • Metric Embedding

Metrics

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

Abstract

We consider the {Requirement Cut} problem, where given an undirected graph G = (V,E) equipped with non-negative edge weights c:E → R_{+}, and g groups of vertices X₁,…,X_{g} ⊆ V each equipped with a requirement r_i, the goal is to find a collection of edges F ⊆ E, with total minimum weight, such that once F is removed from G in the resulting graph every X_{i} is broken into at least r_{i} connected components. {Requirement Cut} captures multiple classic cut problems in graphs, e.g., {Multicut}, {Multiway Cut}, {Min k-Cut}, {Steiner k-Cut}, {Steiner Multicut}, and {Multi-Multiway Cut}. Nagarajan and Ravi [Algoritmica`10] presented an approximation of O(log{n}log{R}) for the problem, which was subsequently improved to O(log{g} log{k}) by Gupta, Nagarajan and Ravi [Operations Research Letters`10] (here R = ∑ _{i = 1}^g r_i and k = |∪ _{i = 1}^g X_i |). We present an approximation of O(Xlog{R} √{log{k}}log{log{k}}) for {Requirement Cut} (here X = max _{i = 1,…,g} {|X_i|}). Our approximation in general is incomparable to the above mentioned previous results, however when all groups are not too large, i.e., X = o((√{log{k}}log{g})/(log{R}log{log{k}})), it is better. Our algorithm is based on a new configuration linear programming relaxation for the problem, which is accompanied by a remarkably simple randomized rounding procedure.

Cite As Get BibTex

Roy Schwartz and Yotam Sharoni. Approximating Requirement Cut via a Configuration LP. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 53:1-53:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.53

Author Details

Roy Schwartz
  • Department of Computer Science, Technion, Haifa, Israel
Yotam Sharoni
  • Department of Computer Science, Technion, Haifa, Israel

Funding

  • Schwartz, Roy: Research is supported by ISF grant 1336/16.

References

  1. Sanjeev Arora, James R. Lee, and Assaf Naor. Fréchet embeddings of negative type metrics. Discret. Comput. Geom., 38(4):726-739, 2007. Google Scholar
  2. Sanjeev Arora, James R. Lee, and Assaf Naor. Euclidean distortion and the sparsest cut. J. Amer. Math. Soc., 21(1):1-21, 2008. Google Scholar
  3. Adi Avidor and Michael Langberg. The multi-multiway cut problem. Theor. Comput. Sci., 377(1-3):35-42, 2007. Google Scholar
  4. Niv Buchbinder, Joseph (Seffi) Naor, and Roy Schwartz. Simplex partitioning via exponential clocks and the multiway-cut problem. SIAM J. Comput., 47(4):1463-1482, 2018. Google Scholar
  5. Niv Buchbinder, Roy Schwartz, and Baruch Weizman. Simplex transformations and the multiway cut problem. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, pages 2400-2410. SIAM, 2017. Google Scholar
  6. Gruia Călinescu, Howard J. Karloff, and Yuval Rabani. An improved approximation algorithm for MULTIWAY CUT. J. Comput. Syst. Sci., 60(3):564-574, 2000. Google Scholar
  7. Shuchi Chawla, Anupam Gupta, and Harald Räcke. Embeddings of negative-type metrics and an improved approximation to generalized sparsest cut. ACM Trans. Algorithms, 4(2):22:1-22:18, 2008. Google Scholar
  8. Chandra Chekuri, Sudipto Guha, and Joseph (Seffi) Naor. The steiner k-cut problem. SIAM J. Discrete Math., 20(1):261-271, 2006. Google Scholar
  9. Elias Dahlhaus, David S. Johnson, Christos H. Papadimitriou, Paul D. Seymour, and Mihalis Yannakakis. The complexity of multiterminal cuts. SIAM J. Comput., 23(4):864-894, 1994. Google Scholar
  10. Michel Marie Deza and Monique Laurent. Geometry of cuts and metrics, volume 15 of Algorithms and combinatorics. Springer, 1997. Google Scholar
  11. Naveen Garg, Vijay V. Vazirani, and Mihalis Yannakakis. Approximate max-flow min-(multi)cut theorems and their applications. SIAM J. Comput., 25(2):235-251, 1996. Google Scholar
  12. Anupam Gupta, Viswanath Nagarajan, and R. Ravi. An improved approximation algorithm for requirement cut. Operations Research Letters, 38(4):322-325, 2010. Google Scholar
  13. David R. Karger, Philip N. Klein, Clifford Stein, Mikkel Thorup, and Neal E. Young. Rounding algorithms for a geometric embedding of minimum multiway cut. Math. Oper. Res., 29(3):436-461, 2004. Google Scholar
  14. Philip N. Klein, Serge A. Plotkin, Satish Rao, and Éva Tardos. Approximation algorithms for steiner and directed multicuts. J. Algorithms, 22(2):241-269, 1997. Google Scholar
  15. Philip N. Klein, Satish Rao, Ajit Agrawal, and R. Ravi. An approximate max-flow min-cut relation for unidirected multicommodity flow, with applications. Combinatorica, 15(2):187-202, 1995. Google Scholar
  16. Frank Thomson Leighton and Satish Rao. Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. J. ACM, 46(6):787-832, 1999. Google Scholar
  17. Nathan Linial, Eran London, and Yuri Rabinovich. The geometry of graphs and some of its algorithmic applications. Combinatorica, 15(2):215-245, 1995. Google Scholar
  18. Viswanath Nagarajan and R. Ravi. Approximation algorithms for requirement cut on graphs. Algorithmica, 56(2):198-213, 2010. Google Scholar
  19. Joseph (Seffi) Naor and Yuval Rabani. Tree packing and approximating k-cuts. In Proceedings of the Twelfth Annual Symposium on Discrete Algorithms, January 7-9, 2001, Washington, DC, USA, pages 26-27, 2001. Google Scholar
  20. R. Ravi and Amitabh Sinha. Approximating k-cuts via network strength. In Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms, January 6-8, 2002, San Francisco, CA, USA, pages 621-622, 2002. Google Scholar
  21. Huzur Saran and Vijay V. Vazirani. Finding k cuts within twice the optimal. SIAM J. Comput., 24(1):101-108, 1995. Google Scholar
  22. Ankit Sharma and Jan Vondrák. Multiway cut, pairwise realizable distributions, and descending thresholds. In David B. Shmoys, editor, Symposium on Theory of Computing, STOC 2014, pages 724-733, 2014. Google Scholar
  23. Neal E. Young. Sequential and parallel algorithms for mixed packing and covering. In 42nd Annual Symposium on Foundations of Computer Science, FOCS 2001, pages 538-546, 2001. 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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact