Document Open Access Logo

Fast Approximation Algorithms for Bounded Degree and Crossing Spanning Tree Problems

Authors Chandra Chekuri, Kent Quanrud, Manuel R. Torres

PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2021.24.pdf
  • Filesize: 0.77 MB
  • 21 pages

Document Identifiers

Author Details

Chandra Chekuri
  • University of Illinois at Urbana-Champaign, IL, USA
Kent Quanrud
  • Purdue University, West Lafayette, IN, USA
Manuel R. Torres
  • University of Illinois at Urbana-Champaign, IL, USA

Funding

  • Chekuri, Chandra: Supported in part by NSF grant CCF-1910149.
  • Torres, Manuel R.: Supported in part by fellowships from NSF and the Sloan Foundation, and NSF grant CCF-1910149.

Cite As Get BibTex

Chandra Chekuri, Kent Quanrud, and Manuel R. Torres. Fast Approximation Algorithms for Bounded Degree and Crossing Spanning Tree Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 24:1-24:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.24

Abstract

We develop fast approximation algorithms for the minimum-cost version of the Bounded-Degree MST problem (BD-MST) and its generalization the Crossing Spanning Tree problem (Crossing-ST). We solve the underlying LP to within a (1+ε) approximation factor in near-linear time via the multiplicative weight update (MWU) technique. This yields, in particular, a near-linear time algorithm that outputs an estimate B such that B ≤ B^* ≤ ⌈(1+ε)B⌉+1 where B^* is the minimum-degree of a spanning tree of a given graph. To round the fractional solution, in our main technical contribution, we describe a fast near-linear time implementation of swap-rounding in the spanning tree polytope of a graph. The fractional solution can also be used to sparsify the input graph that can in turn be used to speed up existing combinatorial algorithms. Together, these ideas lead to significantly faster approximation algorithms than known before for the two problems of interest. In addition, a fast algorithm for swap rounding in the graphic matroid is a generic tool that has other applications, including to TSP and submodular function maximization.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Graph algorithms analysis
Keywords
  • bounded degree spanning tree
  • crossing spanning tree
  • swap rounding
  • fast approximation algorithms

Metrics

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

Related Versions

References

  1. Alexander A. Ageev and Maxim Sviridenko. Pipage rounding: A new method of constructing algorithms with proven performance guarantee. Journal of Combinatorial Optimization, 8:307-328, 2004. Google Scholar
  2. N. Anari and S. O. Gharan. Effective-resistance-reducing flows, spectrally thin trees, and asymmetric TSP. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 20-39, 2015. Google Scholar
  3. Nima Anari, Shayan Oveis Gharan, and Cynthia Vinzant. Log-concave polynomials, entropy, and a deterministic approximation algorithm for counting bases of matroids. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 35-46. IEEE, 2018. Google Scholar
  4. Nima Anari, Kuikui Liu, Shayan Oveis Gharan, and Cynthia Vinzant. Log-concave polynomials IV: exchange properties, tight mixing times, and faster sampling of spanning trees, 2020. URL: http://arxiv.org/abs/2004.07220.
  5. Arash Asadpour, Michel X Goemans, Aleksander Mádry, Shayan Oveis Gharan, and Amin Saberi. An O(log n/log log n)-approximation algorithm for the asymmetric traveling salesman problem. Operations Research, 65(4):1043-1061, 2017. Google Scholar
  6. Nikhil Bansal. On a generalization of iterated and randomized rounding. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 1125-1135, 2019. Google Scholar
  7. Nikhil Bansal, Rohit Khandekar, Jochen Könemann, Viswanath Nagarajan, and Britta Peis. On generalizations of network design problems with degree bounds. Mathematical Programming, 141(1-2):479-506, 2013. Google Scholar
  8. Nikhil Bansal, Rohit Khandekar, and Viswanath Nagarajan. Additive guarantees for degree-bounded directed network design. SIAM Journal on Computing, 39(4):1413-1431, January 2010. URL: https://doi.org/10.1137/080734340.
  9. Vittorio Bilo, Vineet Goyal, Ramamoorthi Ravi, and Mohit Singh. On the crossing spanning tree problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 51-60. Springer, 2004. Google Scholar
  10. Gruia Calinescu, Chandra Chekuri, Martin Pal, and Jan Vondrák. Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing, 40(6):1740-1766, 2011. Google Scholar
  11. Deeparnab Chakrabarty, Yin Tat Lee, Aaron Sidford, Sahil Singla, and Sam Chiu-wai Wong. Faster matroid intersection. In 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS), pages 1146-1168. IEEE, 2019. Google Scholar
  12. Moses Charikar, Alantha Newman, and Aleksandar Nikolov. Tight hardness results for minimizing discrepancy. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms, pages 1607-1614. SIAM, 2011. Google Scholar
  13. Kamalika Chaudhuri, Satish Rao, Samantha Riesenfeld, and Kunal Talwar. What would Edmonds do? augmenting paths and witnesses for degree-bounded MSTs. Algorithmica, 55(1):157-189, November 2007. URL: https://doi.org/10.1007/s00453-007-9115-5.
  14. Chandra Chekuri, Sariel Har-Peled, and Kent Quanrud. Fast LP-based approximations for geometric packing and covering problems. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1019-1038. SIAM, 2020. Google Scholar
  15. Chandra Chekuri and Kent Quanrud. Approximating the Held-Karp bound for metric TSP in nearly-linear time. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 789-800. IEEE, 2017. Google Scholar
  16. Chandra Chekuri and Kent Quanrud. Near-linear time approximation schemes for some implicit fractional packing problems. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 801-820. SIAM, 2017. Google Scholar
  17. Chandra Chekuri and Kent Quanrud. Fast approximations for Metric-TSP via linear programming. arXiv preprint arXiv:1802.01242, 2018. Google Scholar
  18. Chandra Chekuri and Kent Quanrud. Randomized MWU for positive LPs. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 358-377. SIAM, 2018. Google Scholar
  19. Chandra Chekuri, Kent Quanrud, and Manuel R. Torres. Fast approximation algorithms for bounded degree and crossing spanning tree problems. CoRR, abs/2011.03194, 2020. URL: http://arxiv.org/abs/2011.03194.
  20. Chandra Chekuri, Jan Vondrak, and Rico Zenklusen. Dependent randomized rounding via exchange properties of combinatorial structures. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, pages 575-584. IEEE, 2010. Google Scholar
  21. Chandra Chekuri, Jan Vondrák, and Rico Zenklusen. Multi-budgeted matchings and matroid intersection via dependent rounding. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms, pages 1080-1097. SIAM, 2011. Google Scholar
  22. Ran Duan, Haoqing He, and Tianyi Zhang. Near-linear time algorithms for approximate minimum degree spanning trees. ArXiv, abs/1712.09166, 2017. Google Scholar
  23. Alina Ene and Huy L Nguyen. Towards nearly-linear time algorithms for submodular maximization with a matroid constraint. In International Colloquium on Automata, Languages, and Programming, volume 132, 2019. Google Scholar
  24. M. Fürer and B. Raghavachari. Approximating the minimum-degree Steiner tree to within one of optimal. Journal of Algorithms, 17(3):409-423, November 1994. URL: https://doi.org/10.1006/jagm.1994.1042.
  25. Rajiv Gandhi, Samir Khuller, Srinivasan Parthasarathy, and Aravind Srinivasan. Dependent rounding and its applications to approximation algorithms. Journal of the ACM (JACM), 53(3):324-360, 2006. Google Scholar
  26. Kyle Genova and David P Williamson. An experimental evaluation of the best-of-many Christofides' algorithm for the traveling salesman problem. Algorithmica, 78(4):1109-1130, 2017. Google Scholar
  27. Shayan Oveis Gharan, Amin Saberi, and Mohit Singh. A randomized rounding approach to the traveling salesman problem. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pages 550-559. IEEE, 2011. Google Scholar
  28. Michel Goemans. Minimum bounded degree spanning trees. 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS '06), 2006. URL: https://doi.org/10.1109/focs.2006.48.
  29. Jacob Holm, Kristian De Lichtenberg, and Mikkel Thorup. Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. Journal of the ACM (JACM), 48(4):723-760, 2001. Google Scholar
  30. David R Karger. Random sampling and greedy sparsification for matroid optimization problems. Mathematical Programming, 82(1-2):41-81, 1998. Google Scholar
  31. Anna R Karlin, Nathan Klein, and Shayan Oveis Gharan. An improved approximation algorithm for TSP in the half integral case. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 28-39, 2020. Google Scholar
  32. Anna R Karlin, Nathan Klein, and Shayan Oveis Gharan. A (slightly) improved approximation algorithm for metric TSP. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 32-45, 2021. Google Scholar
  33. Tamás Király, Lap Chi Lau, and Mohit Singh. Degree bounded matroids and submodular flows. Combinatorica, 32(6):703-720, 2012. Google Scholar
  34. Jochen Könemann and R Ravi. Primal-dual meets local search: approximating MST’s with nonuniform degree bounds. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 389-395, 2003. Google Scholar
  35. Jochen Könemann and Ramamoorthi Ravi. Primal-dual meets local search: approximating MSTs with nonuniform degree bounds. SIAM Journal on Computing, 34(3):763-773, 2005. Google Scholar
  36. André Linhares and Chaitanya Swamy. Approximating min-cost chain-constrained spanning trees: a reduction from weighted to unweighted problems. Mathematical Programming, 172(1-2):17-34, 2018. Google Scholar
  37. Neil Olver and Rico Zenklusen. Chain-constrained spanning trees. Mathematical Programming, 167(2):293-314, 2018. Google Scholar
  38. Alessandro Panconesi and Aravind Srinivasan. Randomized distributed edge coloring via an extension of the Chernoff-Hoeffding bounds. SIAM J. Comput., 26:350-368, 1997. Google Scholar
  39. Kent Quanrud. Fast and deterministic approximations for k-cut. arXiv preprint arXiv:1807.07143, 2018. Google Scholar
  40. Kent Quanrud. Fast approximations for combinatorial optimization via multiplicative weight updates. PhD thesis, University of Illinois, Urbana-Champaign, 2019. URL: https://www.ideals.illinois.edu/handle/2142/106153.
  41. Aaron Schild. An almost-linear time algorithm for uniform random spanning tree generation. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 214-227, 2018. Google Scholar
  42. Alexander Schrijver. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer Science & Business Media, 2003. Google Scholar
  43. Mohit Singh and Lap Chi Lau. Approximating minimum bounded degree spanning trees to within one of optimal. Journal of the ACM (JACM), 62(1):1-19, 2015. Google Scholar
  44. Mohit Singh and Nisheeth K Vishnoi. Entropy, optimization and counting. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 50-59, 2014. Google Scholar
  45. Aravind Srinivasan. Distributions on level-sets with applications to approximation algorithms. In Proceedings 42nd IEEE Symposium on Foundations of Computer Science, pages 588-597. IEEE, 2001. Google Scholar
  46. Ola Svensson, Jakub Tarnawski, and László A Végh. A constant-factor approximation algorithm for the asymmetric traveling salesman problem. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 204-213, 2018. Google Scholar
  47. Vera Traub and Jens Vygen. An improved approximation algorithm for ATSP. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 1-13, 2020. Google Scholar
  48. Di Wang. Fast Approximation Algorithms for Positive Linear Programs. PhD thesis, EECS Department, University of California, Berkeley, July 2017. URL: http://www2.eecs.berkeley.edu/Pubs/TechRpts/2017/EECS-2017-126.html.
  49. Sein Win. On a connection between the existence of k-trees and the toughness of a graph. Graphs and Combinatorics, 5(1):201-205, 1989. 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
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact