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Lower Bounds on 0-Extension with Steiner Nodes

Authors Yu Chen , Zihan Tan

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Author Details

Yu Chen
  • EPFL, Lausanne, Switzerland
Zihan Tan
  • Rutgers University, Piscataway, NJ, USA

Funding

  • Tan, Zihan: Supported by a grant to DIMACS from the Simons Foundation (820931).

Acknowledgements

We would like to thank Julia Chuzhoy for many helpful discussions. We want to thank Arnold Filtser for pointing to us some previous works on similar problems.

Cite As Get BibTex

Yu Chen and Zihan Tan. Lower Bounds on 0-Extension with Steiner Nodes. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 47:1-47:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.47

Abstract

In the 0-Extension problem, we are given an edge-weighted graph G = (V,E,c), a set T ⊆ V of its vertices called terminals, and a semi-metric D over T, and the goal is to find an assignment f of each non-terminal vertex to a terminal, minimizing the sum, over all edges (u,v) ∈ E, the product of the edge weight c(u,v) and the distance D(f(u),f(v)) between the terminals that u,v are mapped to. Current best approximation algorithms on 0-Extension are based on rounding a linear programming relaxation called the semi-metric LP relaxation. The integrality gap of this LP, is upper bounded by O(log|T|/log log|T|) and lower bounded by Ω((log|T|)^{2/3}), has been shown to be closely related to the quality of cut and flow vertex sparsifiers.
We study a variant of the 0-Extension problem where Steiner vertices are allowed. Specifically, we focus on the integrality gap of the same semi-metric LP relaxation to this new problem. Following from previous work, this new integrality gap turns out to be closely related to the quality achievable by cut/flow vertex sparsifiers with Steiner nodes, a major open problem in graph compression. We show that the new integrality gap stays superconstant Ω(log log |T|) even if we allow a super-linear O(|T|log^{1-ε}|T|) number of Steiner nodes.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sparsification and spanners
Keywords
  • Graph Algorithms
  • Zero Extension
  • Integrality Gap

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References

  1. Alexandr Andoni, Anupam Gupta, and Robert Krauthgamer. Towards (1+ε)-approximate flow sparsifiers. In Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms, pages 279-293. SIAM, 2014. Google Scholar
  2. Haris Angelidakis, Yury Makarychev, and Pasin Manurangsi. An improved integrality gap for the călinescu-karloff-rabani relaxation for multiway cut. In Integer Programming and Combinatorial Optimization: 19th International Conference, IPCO 2017, Waterloo, ON, Canada, June 26-28, 2017, Proceedings, pages 39-50. Springer, 2017. Google Scholar
  3. Aaron Archer, Jittat Fakcharoenphol, Chris Harrelson, Robert Krauthgamer, Kunal Talwar, and Éva Tardos. Approximate classification via earthmover metrics. In Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms, pages 1079-1087. Society for Industrial and Applied Mathematics, 2004. Google Scholar
  4. Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király, and Vivek Madan. Improving the integrality gap for multiway cut. Mathematical Programming, 183(1-2):171-193, 2020. Google Scholar
  5. Niv Buchbinder, Joseph Naor, and Roy Schwartz. Simplex partitioning via exponential clocks and the multiway cut problem. In Proceedings of the forty-fifth annual ACM symposium on Theory of computing, pages 535-544, 2013. Google Scholar
  6. 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, pages 2400-2410. SIAM, 2017. Google Scholar
  7. Gruia Călinescu, Howard Karloff, and Yuval Rabani. An improved approximation algorithm for multiway cut. In Proceedings of the thirtieth annual ACM symposium on Theory of computing, pages 48-52, 1998. Google Scholar
  8. Gruia Calinescu, Howard Karloff, and Yuval Rabani. Approximation algorithms for the 0-extension problem. SIAM Journal on Computing, 34(2):358-372, 2005. Google Scholar
  9. Moses Charikar, Tom Leighton, Shi Li, and Ankur Moitra. Vertex sparsifiers and abstract rounding algorithms. In Foundations of Computer Science (FOCS), 2010 51st Annual IEEE Symposium on, pages 265-274. IEEE, 2010. Google Scholar
  10. Chandra Chekuri, Sanjeev Khanna, Joseph Naor, and Leonid Zosin. A linear programming formulation and approximation algorithms for the metric labeling problem. SIAM Journal on Discrete Mathematics, 18(3):608-625, 2004. Google Scholar
  11. Yu Chen and Zihan Tan. Towards the characterization of terminal cut functions: a condition for laminar families. arXiv preprint, 2023. URL: https://arxiv.org/abs/2310.11367.
  12. Yu Chen and Zihan Tan. On (1+ ε)-approximate flow sparsifiers. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1568-1605. SIAM, 2024. Google Scholar
  13. Julia Chuzhoy. On vertex sparsifiers with steiner nodes. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 673-688, 2012. Google Scholar
  14. Julia Chuzhoy and Joseph Naor. The hardness of metric labeling. SIAM Journal on Computing, 36(5):1376-1386, 2007. Google Scholar
  15. Elias Dahlhaus, David S. Johnson, Christos H. Papadimitriou, Paul D. Seymour, and Mihalis Yannakakis. The complexity of multiterminal cuts. SIAM Journal on Computing, 23(4):864-894, 1994. Google Scholar
  16. Jittat Fakcharoenphol, Chris Harrelson, Satish Rao, and Kunal Talwar. An improved approximation algorithm for the 0-extension problem. In Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms, pages 257-265. Society for Industrial and Applied Mathematics, 2003. Google Scholar
  17. Ari Freund and Howard Karloff. A lower bound of 8/(7+ 1/(k-1)) on the integrality ratio of the călinescu-karloff-rabani relaxation for multiway cut. Information Processing Letters, 75(1-2):43-50, 2000. Google Scholar
  18. Alan Frieze. Hamilton cycles in the union of random permutations. Random Structures & Algorithms, 18(1):83-94, 2001. Google Scholar
  19. Torben Hagerup, Jyrki Katajainen, Naomi Nishimura, and Prabhakar Ragde. Characterizing multiterminal flow networks and computing flows in networks of small treewidth. Journal of Computer and System Sciences, 57(3):366-375, 1998. Google Scholar
  20. Howard Karloff, Subhash Khot, Aranyak Mehta, and Yuval Rabani. On earthmover distance, metric labeling, and 0-extension. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pages 547-556, 2006. Google Scholar
  21. Howard Karloff, Subhash Khot, Aranyak Mehta, and Yuval Rabani. On earthmover distance, metric labeling, and 0-extension. SIAM Journal on Computing, 39(2):371-387, 2009. Google Scholar
  22. Alexander V Karzanov. Minimum 0-extensions of graph metrics. European Journal of Combinatorics, 19(1):71-101, 1998. Google Scholar
  23. Jon Kleinberg and Eva Tardos. Approximation algorithms for classification problems with pairwise relationships: Metric labeling and markov random fields. Journal of the ACM (JACM), 49(5):616-639, 2002. Google Scholar
  24. Robert Krauthgamer and Ron Mosenzon. Exact flow sparsification requires unbounded size. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2354-2367. SIAM, 2023. Google Scholar
  25. William Kuszmaul and Qi Qi. The multiplicative version of azuma’s inequality, with an application to contention analysis. arXiv preprint, 2021. URL: https://arxiv.org/abs/2102.05077.
  26. F Thomson Leighton and Ankur Moitra. Extensions and limits to vertex sparsification. In Proceedings of the forty-second ACM symposium on Theory of computing, pages 47-56. ACM, 2010. Google Scholar
  27. Konstantin Makarychev and Yury Makarychev. Metric extension operators, vertex sparsifiers and lipschitz extendability. In Foundations of Computer Science (FOCS), 2010 51st Annual IEEE Symposium on, pages 255-264. IEEE, 2010. Google Scholar
  28. Rajsekar Manokaran, Joseph Seffi Naor, Prasad Raghavendra, and Roy Schwartz. Sdp gaps and ugc hardness for multiway cut, 0-extension, and metric labeling. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 11-20. ACM, 2008. Google Scholar
  29. Ankur Moitra. Approximation algorithms for multicommodity-type problems with guarantees independent of the graph size. In Foundations of Computer Science, 2009. FOCS'09. 50th Annual IEEE Symposium on, pages 3-12. IEEE, 2009. Google Scholar
  30. Doron Puder. Expansion of random graphs: New proofs, new results. Inventiones mathematicae, 201(3):845-908, 2015. Google Scholar
  31. Roy Schwartz and Nitzan Tur. The metric relaxation for 0-extension admits an ω(log^2/3k) gap. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 1601-1614, 2021. Google Scholar
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