, Volume 6, Issue 1, pp 89–92 | Cite as

Exact and approximate resolution of integral multiflow and multicut problems: algorithms and complexity

  • Cédric BentzEmail author
PhD Thesis


This is a summary of the author’s PhD thesis supervised by Marie- Christine Costa and Frédéric Roupin and defended on 20 November 2006 at the Conservatoire National des Arts et Métiers in Paris (France). The thesis is written in French and is available upon request from the author. This work deals with two well-known optimization problems from graph theory: the maximum integral multiflow and the minimum multicut problems. The main contributions of this thesis concern the polynomial-time solvability and the approximation of these two problems (and of some of their variants) in classical classes of graphs: bounded tree-width graphs, planar graphs and grids, digraphs, etc.


Multicuts Integral multiflows Polynomial-time solvability Polynomial approximation Combinatorial optimization Graph theory 

MSC Classification

05C85 68Q17 90C27 


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  1. Bentz C (2005) Edge disjoint paths and max integral multiflow/min multicut theorems in planar graphs. In: Proceedings ICGT’05, Hyères, Electronic Notes in Discrete Mathematics 22:55–60Google Scholar
  2. Bentz C (2007) The maximum integer multiterminal flow problem in directed graphs. Oper Res Lett 35:195–200CrossRefGoogle Scholar
  3. Bentz C, Costa MC, Picouleau C, Zrikem M (2007a) The shortest multipaths problem in a capacitated dense channel. Eur J Oper Res 178:926–931CrossRefGoogle Scholar
  4. Bentz C, Costa MC, Roupin F (2007b) Maximum integer multiflow and minimum multicut problems in two-sided uniform grid graphs. J Discret algorithms 5:36–54Google Scholar
  5. Chen D, Wu X (2004) Efficient algorithms for k-terminal cuts on planar graphs. Algorithmica 38:299–316CrossRefGoogle Scholar
  6. Costa MC, Létocart L, Roupin F (2005) Minimal multicut and maximal integer multiflow: a survey. Eur J Oper Res 162:55–69CrossRefGoogle Scholar
  7. Dahlhaus E, Johnson D, Papadimitriou C, Seymour P, Yannakakis M (1994) The complexity of multiterminal cuts. SIAM J Comput 23:864–894CrossRefGoogle Scholar
  8. Ford L, Fulkerson D (1956) Maximal flow through a network. Can J Math 8:339–404Google Scholar
  9. Formann M, Wagner D, Wagner F (1993) Routing through a dense channel with minimum total wire length. J Algorithms 15:267–283CrossRefGoogle Scholar
  10. Frank A (1982) Disjoint paths in a rectilinear grid. Combinatorica 2:361–371CrossRefGoogle Scholar
  11. Garg N, Vazirani V, Yannakakis M (1994) Multiway cuts in directed and node weighted graphs. In: Proceedings ICALP, lecture notes in computer science 820:487–498Google Scholar
  12. Garg N, Vazirani V, Yannakakis M (1996) Approximate max-flow min-(multi)cut theorems and their applications. SIAM J Comput 25:235–251CrossRefGoogle Scholar
  13. Garg N, Vazirani V, Yannakakis M (1997) Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18:3–20CrossRefGoogle Scholar
  14. Guruswami V, Khanna S, Rajaraman R, Shepherd B, Yannakakis M (2003) Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems. J Comput Syst Sci 67: 473–496CrossRefGoogle Scholar
  15. Keijsper JCM, Pendavingh RA, Stougie L (2006) A linear programming formulation of Mader’s edge- disjoint paths problem. J Comb Theory Ser B 96:159–163CrossRefGoogle Scholar
  16. Korte B, Lovász L, Prömel HJ, Schrijver A (eds) (1990) Paths, flows and VLSI-layout. Algorithms and combinatorics 9, Springer, BerlinGoogle Scholar
  17. Okamura H, Seymour P (1981) Multicommodity flows in planar graphs. J Comb Theory Ser B 31:75–81CrossRefGoogle Scholar
  18. Tardos E, Vazirani V (1993) Improved bounds for the max-flow min-multicut ratio for planar and K r,r-free graphs. Inform Process Lett 47:77–80CrossRefGoogle Scholar

Copyright information

© Springer Verlag 2007

Authors and Affiliations

  1. 1.CEDRIC LaboratoryConservatoire National des Arts et MétiersParisFrance

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