Papers
Topics
Authors
Recent
Search
2000 character limit reached

On Approximating Cutwidth and Pathwidth

Published 27 Nov 2023 in cs.DS | (2311.15639v2)

Abstract: We study graph ordering problems with a min-max objective. A classical problem of this type is cutwidth, where given a graph we want to order its vertices such that the number of edges crossing any point is minimized. We give a $ \log{1+o(1)}(n)$ approximation for the problem, substantially improving upon the previous poly-logarithmic guarantees based on the standard recursive balanced partitioning approach of Leighton and Rao (FOCS'88). Our key idea is a new metric decomposition procedure that is suitable for handling min-max objectives, which could be of independent interest. We also use this to show other results, including an improved $ \log{1+o(1)}(n)$ approximation for computing the pathwidth of a graph.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (59)
  1. o⁢(log⁡n)𝑜𝑛o(\sqrt{\log n})italic_o ( square-root start_ARG roman_log italic_n end_ARG ) approximation algorithms for min uncut, min 2cnf deletion, and directed cut problems. In Symposium on Theory of Computing (STOC), pages 573–581, 2005.
  2. A new rounding procedure for the assignment problem with applications to dense graph arrangement problems. Math. Program., 92(1):1–36, 2002.
  3. Fréchet embeddings of negative type metrics. Discret. Comput. Geom., 38(4):726–739, 2007.
  4. Sparse partitions. In Symposium on Foundations of Computer Science, 1990.
  5. Expander flows, geometric embeddings and graph partitioning. Journal of the ACM (JACM), 56(2):1–37, 2009.
  6. Yair Bartal. Probabilistic approximation of metric spaces and its algorithmic applications. In Foundations of Computer Science, pages 184–193, 1996.
  7. Yair Bartal. On approximating arbitrary metrices by tree metrics. In Symposium on Theory of Computing, STOC, page 161–168, 1998.
  8. Fixed-parameter algorithms for protein similarity search under mRNA structure constraints. Journal of Discrete Algorithms, 6(4):618–626, 2008.
  9. A note on exact algorithms for vertex ordering problems on graphs. Theory Comput. Syst., 50(3):420–432, 2012.
  10. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. J. Algorithms, 18(2):238–255, 1995.
  11. Linear-time register allocation for a fixed number of registers. In Symposium on Discrete Algorithms, SODA, page 574–583, 1998.
  12. Efficient and constructive algorithms for the pathwidth and treewidth of graphs. Journal of Algorithms, 21(2):358–402, 1996.
  13. Grundy distinguishes treewidth from pathwidth. SIAM Journal on Discrete Mathematics, 36(3):1761–1787, 2022.
  14. Semi-definite relaxations for minimum bandwidth and other vertex-ordering problems. Theor. Comput. Sci., 235(1):25–42, 2000.
  15. A framework for solving VLSI graph layout problems. J. Comput. Syst. Sci., 28(2):300–343, 1984.
  16. Hans L. Bodlaender. A tourist guide through treewidth. Acta Cybern., 11(1-2):1–21, 1993.
  17. Hans L. Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theoretical Computer Science, 209(1):1–45, 1998.
  18. Flow metrics. Theoretical Computer Science, 321(1):13–24, 2004. Latin American Theoretical Informatics.
  19. Graph and String Parameters: Connections Between Pathwidth, Cutwidth and the Locality Number. In International Colloquium on Automata, Languages, and Programming (ICALP), pages 109:1–109:16, 2019.
  20. ℓ22superscriptsubscriptℓ22\ell_{2}^{2}roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT spreading metrics for vertex ordering problems. Algorithmica, 56(4):577–604, 2010.
  21. On cutwidth parameterized by vertex cover. Algorithmica, 68:940–953, 2014.
  22. Divide-and-conquer approximation algorithms via spreading metrics. Journal of the ACM (JACM), 47(4):585–616, 2000.
  23. Graph separation and search number. Technical Report, 1987.
  24. Uriel Feige. Approximating the bandwidth via volume respecting embeddings. Journal of Computer and System Sciences, 60(3):510–539, 2000.
  25. Improved approximation algorithms for minimum-weight vertex separators. In Symposium on Theory of Computing, STOC, pages 563–572, 2005.
  26. Exact Exponential Algorithms. Springer-Verlag, Berlin, Heidelberg, 2010.
  27. On search, decision, and the efficiency of polynomial-time algorithms. Journal of Computer and System Sciences, 49(3):769–779, 1994.
  28. An improved approximation ratio for the minimum linear arrangement problem. Information Processing Letters, 101(1):26–29, 2007.
  29. A tight bound on approximating arbitrary metrics by tree metrics. J. Comput. Syst. Sci., 69(3):485–497, 2004.
  30. Approximating pathwidth for graphs of small treewidth. In Symposium on Discrete Algorithms (SODA), pages 1965–1976, 2021.
  31. Cutwidth: obstructions and algorithmic aspects. Algorithmica, 81:557–588, 2019.
  32. Partitioning into expanders. In Chandra Chekuri, editor, Symposium on Discrete Algorithms, SODA, pages 1256–1266, 2014.
  33. Anupam Gupta. Improved bandwidth approximation for trees and chordal graphs. J. Algorithms, 40(1):24–36, 2001.
  34. M. D. Hansen. Approximation algorithms for geometric embeddings in the plane with applications to parallel processing problems. In Foundations of Computer Science (FOCS), pages 604–609, 1989.
  35. Approximation through multicommodity flow. In Foundations of Computer Science, FOCS, pages 726–737, 1990.
  36. Nancy G Kinnersley. The vertex separation number of a graph equals its path-width. Information Processing Letters, 42(6):345–350, 1992.
  37. Measured descent: A new embedding method for finite metrics. Geometric and Functional Analysis, 15(4):839–858, August 2005.
  38. Scheduling series-parallel task graphs to minimize peak memory. Theoretical Computer Science, 707:1–23, 2018.
  39. Tree-width, path-width, and cutwidth. Discrete Applied Mathematics, 43(1):97–101, 1993.
  40. Narrowness, pathwidth, and their application in natural language processing. Discrete Applied Mathematics, 36(1):87–92, 1992.
  41. On clusterings: Good, bad and spectral. J. ACM, 51(3):497–515, may 2004.
  42. Thomas Lengauer. Black-white pebbles and graph separation. Acta Informatica, 16, 1981.
  43. Joseph WH Liu. An application of generalized tree pebbling to sparse matrix factorization. SIAM Journal on Algebraic Discrete Methods, 8(3):375–395, 1987.
  44. T. Leighton and S. Rao. An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms. In Foundations of Computer Science, FOCS, pages 422–431, 1988.
  45. Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. J. ACM, 46(6):787–832, November 1999.
  46. Rolf H. Möhring. Graph Problems Related to Gate Matrix Layout and PLA Folding, pages 17–51. Springer Vienna, 1990.
  47. B. Monien and I.H. Sudborough. Min cut is np-complete for edge weighted trees. Theoretical Computer Science, 58(1):209–229, 1988.
  48. On minimizing width in linear layouts. Discrete Applied Mathematics, 23(3):243–265, 1989.
  49. Harald Räcke. Optimal hierarchical decompositions for congestion minimization in networks. In Symposium on Theory of computing, STOC, pages 255–264, 2008.
  50. Harald Räcke. Survey on oblivious routing strategies. In Klaus Ambos-Spies, Benedikt Löwe, and Wolfgang Merkle, editors, Mathematical Theory and Computational Practice, pages 419–429. Springer, 2009.
  51. Ordering problems approximated: single-processor scheduling and interval graph completion. In ICALP, pages 751–762, 1991.
  52. New approximation techniques for some linear ordering problems. SIAM Journal on Computing, 34(2):388–404, 2005.
  53. Graph minors. i. excluding a forest. Journal of Combinatorial Theory, Series B, 35(1):39–61, 1983.
  54. Paul D. Seymour. Packing directed circuits fractionally. Combinatorica, 15(2):281–288, 1995.
  55. Spectral sparsification of graphs. SIAM Journal on Computing, 40(4):981–1025, 2011.
  56. Cutwidth i: A linear time fixed parameter algorithm. Journal of Algorithms, 56(1):1–24, 2005.
  57. Cutwidth ii: Algorithms for partial w-trees of bounded degree. Journal of Algorithms, 56(1):25–49, 2005.
  58. Inapproximability of treewidth, one-shot pebbling, and related layout problems. J. Artif. Int. Res., 49(1):569–600, jan 2014.
  59. Mihalis Yannakakis. A polynomial algorithm for the min-cut linear arrangement of trees. Journal of the ACM (JACM), 32(4):950–988, 1985.

Summary

No one has generated a summary of this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Sign Up to Summarize

Paper to Video (Beta)

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up

Tweets

Sign up for free to view the 1 tweet with 0 likes about this paper.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up for Free