Skip to main content
SpringerLink
Log in

Discrete Convex Functions on Graphs and Their Algorithmic Applications

  • Chapter
  • First Online:
Combinatorial Optimization and Graph Algorithms

Abstract

The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by combinatorial dualities in multiflow problems and the complexity classification of facility location problems on graphs. We outline the theory and algorithmic applications in combinatorial optimization problems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. P. Abramenko, K.S. Brown, Buildings-Theory and Applications (Springer, New York, 2008)

    MATH  Google Scholar 

  2. E. Anshelevich, A. Karagiozova, Terminal backup, 3D matching, and covering cubic graphs. SIAM J. Comput. 40, 678–708 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. M.A. Babenko, A fast algorithm for the path 2-packing problem. Theory Comput. Syst. 46, 59–79 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. M.A. Babenko, A.V. Karzanov, A scaling algorithm for the maximum node-capacitated multiflow problem, in Proceedings of 16th Annual European Symposium on Algorithms (ESA’08). LNCS, vol. 5193 (2008), pp. 124–135

    Google Scholar 

  5. H.-J. Bandelt, M. van de Vel, E. Verheul, Modular interval spaces. Math. Nachr. 163, 177–201 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Bernáth, Y. Kobayashi, T. Matsuoka, The generalized terminal backup problem. SIAM J. Discret. Math. 29, 1764–1782 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. M.R. Bridson, A. Haefliger, Metric Spaces of Non-positive Curvature (Springer, Berlin, 1999)

    Book  MATH  Google Scholar 

  8. J. Chalopin, V. Chepoi, H. Hirai, D. Osajda, Weakly modular graphs and nonpositive curvature. preprint (2014), arXiv:1409.3892

  9. V. Chepoi, A multifacility location problem on median spaces. Discret. Appl. Math. 64, 1–29 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Favati, F. Tardella, Convexity in nonlinear integer programming. Ric. Oper. 53, 3–44 (1990)

    Google Scholar 

  11. A. Frank, Connections in Combinatorial Optimization (Oxford University Press, Oxford, 2011)

    MATH  Google Scholar 

  12. S. Fujishige, Submodular Functions and Optimization, 2nd edn. (Elsevier, Amsterdam, 2005)

    MATH  Google Scholar 

  13. S. Fujishige, Bisubmodular polyhedra, simplicial divisions, and discrete convexity. Discret. Optim. 12, 115–120 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. S. Fujishige, S. Iwata, Algorithms for submodular flows. IEICE Trans. Inf. Syst. 83, 322–329 (2000)

    Google Scholar 

  15. S. Fujishige, T. Király, K. Makino, K. Takazawa, S. Tanigawa, Minimizing submodular functions on diamonds via generalized fractional matroid matchings. EGRES Technical Report (TR-2014-14) (2014)

    Google Scholar 

  16. S. Fujishige, K. Murota, Notes on L-/M-convex functions and the separation theorems. Math. Progr. Ser. A 88, 129–146 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Fujishige, X. Zhang, New algorithms for the intersection problem of submodular systems. Japan J. Ind. Appl. Math. 9, 369–382 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. T. Fukunaga, Approximating the generalized terminal backup problem via half-integral multiflow relaxation. SIAM J. Discret. Math. 30, 777–800 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. N. Garg, V.V. Vazirani, M. Yannakakis, Multiway cuts in node weighted graphs. J. Algorithms 50, 49–61 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. A.V. Goldberg, A.V. Karzanov, Scaling methods for finding a maximum free multiflow of minimum cost. Math. Oper. Res. 22, 90–109 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. G. Grätzer, Lattice Theory: Foundation (Birkhäuser, Basel, 2011)

    Book  MATH  Google Scholar 

  22. R. Hassin, The minimum cost flow problem: a unifying approach to dual algorithms and a new tree-search algorithm. Math. Progr. 25, 228–239 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  23. H. Hirai, Folder complexes and multiflow combinatorial dualities. SIAM J. Discret. Math. 25, 1119–1143 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. H. Hirai, Half-integrality of node-capacitated multiflows and tree-shaped facility locations on trees. Math. Progr. Ser. A 137, 503–530 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. H. Hirai, L-extendable functions and a proximity scaling algorithm for minimum cost multiflow problem. Discret. Optim. 18, 1–37 (2015)

    Article  MathSciNet  Google Scholar 

  26. H. Hirai, A dual descent algorithm for node-capacitated multiflow problems and its applications. preprint (2015), arXiv:1508.07065

  27. H. Hirai, Discrete convexity and polynomial solvability in minimum 0-extension problems. Math. Progr. Ser. A 155, 1–55 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. H. Hirai, L-convexity on graph structures (2016), arXiv:1610.02469

  29. A. Huber, V. Kolmogorov, Towards minimizing \(k\)-submodular functions. in Proceedings of the 2nd International Symposium on Combinatorial Optimization (ISCO’12). LNCS, vol. 7422 (Springer, Berlin, 2012), pp. 451–462

    Google Scholar 

  30. Y. Iwamasa, On a general framework for network representability in discrete optimization. J. Comb. Optim. (to appear)

    Google Scholar 

  31. Y. Iwata, M. Wahlström, Y. Yoshida, Half-integrality, LP-branching and FPT algorithms. SIAM J. Comput. 45, 1377–1411 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. A.V. Karzanov, A minimum cost maximum multiflow problem, in Combinatorial Methods for Flow Problems, Institute for System Studies (Moscow, 1979), pp. 138–156 (Russian)

    Google Scholar 

  33. A.V. Karzanov, Minimum cost multiflows in undirected networks. Math. Progr. Ser. A 66, 313–324 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  34. A.V. Karzanov, Minimum \(0\)-extensions of graph metrics. Eur. J. Comb. 19, 71–101 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. A.V. Karzanov, One more well-solved case of the multifacility location problem. Discret. Optim. 1, 51–66 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  36. J. Kleinberg, É. Tardos, Approximation algorithms for classification problems with pairwise relationships: metric labeling and Markov random fields. J. ACM 49, 616–639 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  37. A.W.J. Kolen, Tree Network and Planar Rectilinear Location Theory, CWI Tract 25 (Center for Mathematics and Computer Science, Amsterdam, 1986)

    MATH  Google Scholar 

  38. V. Kolmogorov, Submodularity on a tree: unifying L\(^\natural \)-convex and bisubmodular functions. in Proceedings of the 36th International Symposium on Mathematical Foundations of Computer Science (MFCS’11). LNCS, vol. 6907 (Springer, Berlin, 2011), pp. 400–411

    Google Scholar 

  39. V. Kolmogorov, J. Thapper, S. Živný, The power of linear programming for general-valued CSPs. SIAM J. Comput. 44, 1–36 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  40. F. Kuivinen, Algorithms and hardness results for some valued CSPs, dissertation No. 1274, Linköping Studies in Science and Technology, Linköping University, Linköping Sweden (2009)

    Google Scholar 

  41. F. Kuivinen, On the complexity of submodular function minimisation on diamonds. Discret. Optim. 8, 459–477 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  42. L. Lovász, Submodular functions and convexity. in eds. By A. Bachem, M. Grötschel, B. Korte, Mathematical Programming—The State of the Art (Springer, Berlin, 1983), pp. 235–257

    Google Scholar 

  43. K. Murota, Discrete convex analysis. Math. Progr. 83, 313–371 (1998)

    MathSciNet  MATH  Google Scholar 

  44. K. Murota, Discrete Convex Analysis (SIAM, Philadelphia, 2003)

    Book  MATH  Google Scholar 

  45. K. Murota, Recent developments in discrete convex analysis, in eds. By W.J. Cook, L. Lovász, J. Vygen, Research Trends in Combinatorial Optimization (Springer, Berlin, 2009), pp. 219–260

    Google Scholar 

  46. K. Murota, A. Shioura, Exact bounds for steepest descent algorithms of L-convex function minimization. Oper. Res. Lett. 42, 361–366 (2014)

    Article  MathSciNet  Google Scholar 

  47. G. Pap, Some new results on node-capacitated packing of A-paths, in Proceedings of the 39th Annual ACM Symposium on Theory of Computing (STOC’07) (ACM, New York, 2007), pp. 599–604

    Google Scholar 

  48. G. Pap, Strongly polynomial time solvability of integral and half-integral node-capacitate multiflow problems, EGRES Technical Report, TR-2008-12 (2008)

    Google Scholar 

  49. A. Shioura, Algorithms for L-convex function minimization: connection between discrete convex analysis and other research fields. J. Oper. Res. Soc. Japan. (to appear)

    Google Scholar 

  50. B.C. Tansel, R.L. Francis, T.J. Lowe, Location on networks I. II Manag. Sci. 29, 498–511 (1983)

    Article  MATH  Google Scholar 

  51. J. Thapper, S. Živný, The complexity of finite-valued CSPs. J. ACM 63(37) (2016)

    Google Scholar 

  52. J. Tits Buildings of Spherical Type and Finite BN-pairs. Lecture Notes in Mathematics, vol. 386 (Springer, New York, 1974)

    Google Scholar 

  53. V.V. Vazirani, Approximation Algorithms (Springer, Berlin, 2001)

    MATH  Google Scholar 

  54. S. Živný, The Complexity of Valued Constraint Satisfaction Problems (Springer, Heidelberg, 2012)

    MATH  Google Scholar 

Download references

Acknowledgements

The author thanks Yuni Iwamasa for careful reading, Satoru Fujishige for remarks, and Kazuo Murota for numerous comments improving presentation. The work was partially supported by JSPS KAKENHI Grant Numbers 25280004, 26330023, 26280004, 17K00029.

Author information

Authors and Affiliations

  1. Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo, 113-8656, Japan

    Hiroshi Hirai

Authors
  1. Hiroshi Hirai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hiroshi Hirai .

Editor information

Editors and Affiliations

  1. National Institute of Informatics , Tokyo, Japan

    Takuro Fukunaga

  2. National Institute of Informatics , Tokyo, Japan

    Ken-ichi Kawarabayashi

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer Nature Singapore Pte Ltd.

About this chapter

Cite this chapter

Hirai, H. (2017). Discrete Convex Functions on Graphs and Their Algorithmic Applications. In: Fukunaga, T., Kawarabayashi, Ki. (eds) Combinatorial Optimization and Graph Algorithms. Springer, Singapore. https://doi.org/10.1007/978-981-10-6147-9_4

Download citation

  • DOI: https://doi.org/10.1007/978-981-10-6147-9_4

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-6146-2

  • Online ISBN: 978-981-10-6147-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation