Graph and String Parameters: Connections Between Pathwidth, Cutwidth and the Locality Number (Track B: Automata, Logic, Semantics, and Theory of Programming)

Authors Katrin Casel, Joel D. Day , Pamela Fleischmann , Tomasz Kociumaka , Florin Manea , Markus L. Schmid



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2019.109.pdf
  • Filesize: 0.6 MB
  • 16 pages

Document Identifiers

Author Details

Katrin Casel
  • Hasso Plattner Institute, University of Potsdam, Germany
Joel D. Day
  • Department of Computer Science, Loughborough University, UK
Pamela Fleischmann
  • Department of Computer Science, Kiel University, Germany
Tomasz Kociumaka
  • Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel
  • Institute of Informatics, University of Warsaw, Poland
Florin Manea
  • Department of Computer Science, Kiel University, Germany
Markus L. Schmid
  • Trier University, Germany

Cite AsGet BibTex

Katrin Casel, Joel D. Day, Pamela Fleischmann, Tomasz Kociumaka, Florin Manea, and Markus L. Schmid. Graph and String Parameters: Connections Between Pathwidth, Cutwidth and the Locality Number (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 109:1-109:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ICALP.2019.109

Abstract

We investigate the locality number, a recently introduced structural parameter for strings (with applications in pattern matching with variables), and its connection to two important graph-parameters, cutwidth and pathwidth. These connections allow us to show that computing the locality number is NP-hard but fixed-parameter tractable (when the locality number or the alphabet size is treated as a parameter), and can be approximated with ratio O(sqrt{log{opt}} log n). As a by-product, we also relate cutwidth via the locality number to pathwidth, which is of independent interest, since it improves the best currently known approximation algorithm for cutwidth. In addition to these main results, we also consider the possibility of greedy-based approximation algorithms for the locality number.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Combinatorics on words
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Graph and String Parameters
  • NP-Completeness
  • Approximation Algorithms

Metrics

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

References

  1. Amihood Amir and Igor Nor. Generalized function matching. Journal of Discrete Algorithms, 5(3):514-523, 2007. URL: http://dx.doi.org/10.1016/j.jda.2006.10.001.
  2. Dana Angluin. Finding patterns common to a set of strings. Journal of Computer and System Sciences, 21(1):46-62, 1980. URL: http://dx.doi.org/10.1016/0022-0000(80)90041-0.
  3. Sanjeev Arora, Boaz Barak, and David Steurer. Subexponential Algorithms for Unique Games and Related Problems. Journal of the ACM, 62(5):42:1-42:25, 2015. URL: http://dx.doi.org/10.1145/2775105.
  4. Sanjeev Arora, Satish Rao, and Umesh V. Vazirani. Expander flows, geometric embeddings and graph partitioning. Journal of the ACM, 56(2):5:1-5:37, 2009. URL: http://dx.doi.org/10.1145/1502793.1502794.
  5. Giorgio Ausiello, Alberto Marchetti-Spaccamela, Pierluigi Crescenzi, Giorgio Gambosi, Marco Protasi, and Viggo Kann. Complexity and approximation: combinatorial optimization problems and their approximability properties. Springer, 1999. URL: http://dx.doi.org/10.1007/978-3-642-58412-1.
  6. Boaz Barak, Prasad Raghavendra, and David Steurer. Rounding Semidefinite Programming Hierarchies via Global Correlation. In Rafail Ostrovsky, editor, 52nd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2011, pages 472-481. IEEE, 2011. URL: http://dx.doi.org/10.1109/focs.2011.95.
  7. Pablo Barceló, Leonid Libkin, Anthony W. Lin, and Peter T. Wood. Expressive Languages for Path Queries over Graph-Structured Data. ACM Transactions on Database Systems, 37(4):1-46, 2012. URL: http://dx.doi.org/10.1145/2389241.2389250.
  8. Hans L. Bodlaender. A Tourist Guide through Treewidth. Acta Cybernetica, 11(1-2):1-21, 1993. URL: http://www.inf.u-szeged.hu/actacybernetica/edb/vol11n1_2/pdf/Bodlaender_1993_ActaCybernetica.pdf.
  9. Hans L. Bodlaender. A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth. SIAM Journal on Computing, 25(5):1305-1317, 1996. URL: http://dx.doi.org/10.1137/s0097539793251219.
  10. Hans L. Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theoretical Computer Science, 209(1-2):1-45, 1998. URL: http://dx.doi.org/10.1016/S0304-3975(97)00228-4.
  11. Hans L. Bodlaender. Fixed-Parameter Tractability of Treewidth and Pathwidth. In Hans L. Bodlaender, Rod Downey, Fedor V. Fomin, and Dániel Marx, editors, The Multivariate Algorithmic Revolution and Beyond, volume 7370 of LNCS, pages 196-227, 2012. URL: http://dx.doi.org/10.1007/978-3-642-30891-8_12.
  12. Hans L. Bodlaender, Fedor V. Fomin, Arie M. C. A. Koster, Dieter Kratsch, and Dimitrios M. Thilikos. A Note on Exact Algorithms for Vertex Ordering Problems on Graphs. Theory of Computing Systems, 50(3):420-432, 2012. URL: http://dx.doi.org/10.1007/s00224-011-9312-0.
  13. David Coudert, Dorian Mazauric, and Nicolas Nisse. Experimental Evaluation of a Branch-and-Bound Algorithm for Computing Pathwidth and Directed Pathwidth. ACM Journal of Experimental Algorithmics, 21(1):1.3:1-1.3:23, 2016. URL: http://dx.doi.org/10.1145/2851494.
  14. Joel D. Day, Pamela Fleischmann, Florin Manea, and Dirk Nowotka. Local Patterns. In Satya V. Lokam and R. Ramanujam, editors, Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2017, volume 93 of LIPIcs, pages 24:1-24:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2017.24.
  15. Joel D. Day, Pamela Fleischmann, Florin Manea, Dirk Nowotka, and Markus L. Schmid. On Matching Generalised Repetitive Patterns. In Mizuho Hoshi and Shinnosuke Seki, editors, Developments in Language Theory, DLT 2018, volume 11088 of LNCS, pages 269-281. Springer, 2018. URL: http://dx.doi.org/10.1007/978-3-319-98654-8_22.
  16. Holger Dell, Christian Komusiewicz, Nimrod Talmon, and Mathias Weller. The PACE 2017 Parameterized Algorithms and Computational Experiments Challenge: The Second Iteration. In Daniel Lokshtanov and Naomi Nishimura, editors, Parameterized and Exact Computation, IPEC 2017, volume 89 of LIPIcs, pages 30:1-30:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.IPEC.2017.30.
  17. Josep Díaz, Jordi Petit, and Maria Serna. A Survey of Graph Layout Problems. ACM Computing Surveys, 34(3):313-356, 2002. URL: http://dx.doi.org/10.1145/568522.568523.
  18. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Springer, 2013. URL: http://dx.doi.org/10.1007/978-1-4471-5559-1.
  19. Thomas Erlebach, Peter Rossmanith, Hans Stadtherr, Angelika Steger, and Thomas Zeugmann. Learning one-variable pattern languages very efficiently on average, in parallel, and by asking queries. Theoretical Computer Science, 261(1):119-156, 2001. URL: http://dx.doi.org/10.1016/s0304-3975(00)00136-5.
  20. Uriel Feige, MohammadTaghi HajiAghayi, and James R. Lee. Improved Approximation Algorithms for Minimum Weight Vertex Separators. SIAM Journal on Computing, 38(2):629-657, 2008. URL: http://dx.doi.org/10.1137/05064299x.
  21. Henning Fernau, Florin Manea, Robert Mercas, and Markus L. Schmid. Pattern Matching with Variables: Fast Algorithms and New Hardness Results. In Ernst W. Mayr and Nicolas Ollinger, editors, Symposium on Theoretical Aspects of Computer Science, STACS 2015, volume 30 of LIPIcs, pages 302-315. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2015.302.
  22. Henning Fernau, Florin Manea, Robert Mercaş, and Markus L. Schmid. Revisiting Shinohara’s Algorithm for Computing Descriptive Patterns. Theoretical Computer Science, 733:44-54, 2018. URL: http://dx.doi.org/10.1016/j.tcs.2018.04.035.
  23. Henning Fernau and Markus L. Schmid. Pattern matching with variables: A multivariate complexity analysis. Information and Computation, 242:287-305, 2015. URL: http://dx.doi.org/10.1016/j.ic.2015.03.006.
  24. Henning Fernau, Markus L. Schmid, and Yngve Villanger. On the Parameterised Complexity of String Morphism Problems. Theory of Computing Systems, 59(1):24-51, 2016. URL: http://dx.doi.org/10.1007/s00224-015-9635-3.
  25. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Springer, 2006. URL: http://dx.doi.org/10.1007/3-540-29953-X.
  26. Dominik D. Freydenberger. Extended Regular Expressions: Succinctness and Decidability. Theory of Computing Systems, 53(2):159-193, 2013. URL: http://dx.doi.org/10.1007/s00224-012-9389-0.
  27. Dominik D. Freydenberger and Markus L. Schmid. Deterministic regular expressions with back-references. Journal of Computer and System Sciences, 2019. URL: http://dx.doi.org/10.1016/j.jcss.2019.04.001.
  28. Jeffrey E. F. Friedl. Mastering Regular Expressions. O'Reilly, Sebastopol, CA, 3rd edition, 2006. Google Scholar
  29. Venkatesan Guruswami and Ali Kemal Sinop. Lasserre Hierarchy, Higher Eigenvalues, and Approximation Schemes for Graph Partitioning and Quadratic Integer Programming with PSD Objectives. In Rafail Ostrovsky, editor, 52nd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2011, pages 482-491. IEEE, 2011. URL: http://dx.doi.org/10.1109/FOCS.2011.36.
  30. Juhani Karhumäki, Filippo Mignosi, and Wojciech Plandowski. The expressibility of languages and relations by word equations. Journal of the ACM, 47(3):483-505, 2000. URL: http://dx.doi.org/10.1145/337244.337255.
  31. Michael Kearns and Leonard Pitt. A polynomial-time algorithm for learning k-variable pattern languages from examples. In Ronald L. Rivest, David Haussler, and Manfred K. Warmuth, editors, Computational Learning Theory, COLT 1989, pages 57-71. Morgan Kaufmann, 1989. URL: http://dx.doi.org/10.1016/b978-0-08-094829-4.50007-6.
  32. Subhash Khot. On the power of unique 2-prover 1-round games. In John H. Reif, editor, 34th Annual ACM Symposium on Theory of Computing, STOC 2002, pages 767-775. ACM, 2002. URL: http://dx.doi.org/10.1145/509907.510017.
  33. Subhash Khot. On the Unique Games Conjecture (Invited Survey). In Computational Complexity, CCC 2010, pages 99-121. IEEE, 2010. URL: http://dx.doi.org/10.1109/CCC.2010.19.
  34. Ton Kloks, editor. Treewidth, Computations and Approximations, volume 842 of LNCS. Springer, 1994. URL: http://dx.doi.org/10.1007/BFb0045375.
  35. Tom Leighton and Satish Rao. Multicommodity Max-flow Min-cut Theorems and Their Use in Designing Approximation Algorithms. Journal of the ACM, 46(6):787-832, 1999. URL: http://dx.doi.org/10.1145/331524.331526.
  36. M. Lothaire, editor. Algebraic Combinatorics on Words. Cambridge University Press, 2002. URL: http://dx.doi.org/10.1017/cbo9781107326019.
  37. Fillia Makedon, Christos H. Papadimitriou, and Ivan Hal Sudborough. Topological Bandwidth. SIAM Journal on Algebraic and Discrete Methods, 6(3):418-444, 1985. URL: http://dx.doi.org/10.1137/0606044.
  38. Yen Kaow Ng and Takeshi Shinohara. Developments from enquiries into the learnability of the pattern languages from positive data. Theoretical Computer Science, 397(1-3):150-165, 2008. URL: http://dx.doi.org/10.1016/j.tcs.2008.02.028.
  39. Jordi Petit. Addenda to the Survey of Layout Problems. Bulletin of the EATCS, 105:177-201, 2011. URL: http://eatcs.org/beatcs/index.php/beatcs/article/view/98.
  40. Prasad Raghavendra and David Steurer. Graph expansion and the unique games conjecture. In Leonard J. Schulman, editor, 42nd ACM Symposium on Theory of Computing, STOC 2010, pages 755-764. ACM, 2010. URL: http://dx.doi.org/10.1145/1806689.1806792.
  41. Prasad Raghavendra, David Steurer, and Madhur Tulsiani. Reductions between Expansion Problems. In Computational Complexity, CCC 2012, pages 64-73. IEEE, 2012. URL: http://dx.doi.org/10.1109/CCC.2012.43.
  42. Daniel Reidenbach. Discontinuities in pattern inference. Theoretical Computer Science, 397(1-3):166-193, 2008. URL: http://dx.doi.org/10.1016/j.tcs.2008.02.029.
  43. Daniel Reidenbach and Markus L. Schmid. Patterns with bounded Treewidth. Information and Computation, 239:87-99, 2014. URL: http://dx.doi.org/10.1016/j.ic.2014.08.010.
  44. Markus L. Schmid. Characterising REGEX languages by regular languages equipped with factor-referencing. Information and Computation, 249:1-17, 2016. URL: http://dx.doi.org/10.1016/j.ic.2016.02.003.
  45. Takeshi Shinohara. Polynomial Time Inference of Pattern Languages and Its Application. In 7th IBM Symposium on Mathematical Foundations of Computer Science, pages 191-209, 1982. Google Scholar
  46. Karol Suchan and Yngve Villanger. Computing Pathwidth Faster Than 2ⁿ. In Jianer Chen and Fedor V. Fomin, editors, Parameterized and Exact Computation, IWPEC 2009, volume 5917 of LNCS, pages 324-335. Springer, 2009. URL: http://dx.doi.org/10.1007/978-3-642-11269-0_27.
  47. Dimitrios M. Thilikos, Maria J. Serna, and Hans L. Bodlaender. Cutwidth I: A linear time fixed parameter algorithm. Journal of Algorithms, 56(1):1-24, 2005. URL: http://dx.doi.org/10.1016/j.jalgor.2004.12.001.
  48. Yu (Ledell) Wu, Per Austrin, Toniann Pitassi, and David Liu. Inapproximability of Treewidth and Related Problems. Journal of Artificial Intelligence Research, 49(1):569-600, 2014. URL: http://dx.doi.org/10.1613/jair.4030.
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