Abstract
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.
Similar content being viewed by others
References
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 22:55–60
Bentz C (2007) The maximum integer multiterminal flow problem in directed graphs. Oper Res Lett 35:195–200
Bentz C, Costa MC, Picouleau C, Zrikem M (2007a) The shortest multipaths problem in a capacitated dense channel. Eur J Oper Res 178:926–931
Bentz C, Costa MC, Roupin F (2007b) Maximum integer multiflow and minimum multicut problems in two-sided uniform grid graphs. J Discret algorithms 5:36–54
Chen D, Wu X (2004) Efficient algorithms for k-terminal cuts on planar graphs. Algorithmica 38:299–316
Costa MC, Létocart L, Roupin F (2005) Minimal multicut and maximal integer multiflow: a survey. Eur J Oper Res 162:55–69
Dahlhaus E, Johnson D, Papadimitriou C, Seymour P, Yannakakis M (1994) The complexity of multiterminal cuts. SIAM J Comput 23:864–894
Ford L, Fulkerson D (1956) Maximal flow through a network. Can J Math 8:339–404
Formann M, Wagner D, Wagner F (1993) Routing through a dense channel with minimum total wire length. J Algorithms 15:267–283
Frank A (1982) Disjoint paths in a rectilinear grid. Combinatorica 2:361–371
Garg N, Vazirani V, Yannakakis M (1994) Multiway cuts in directed and node weighted graphs. In: Proceedings ICALP, lecture notes in computer science 820:487–498
Garg N, Vazirani V, Yannakakis M (1996) Approximate max-flow min-(multi)cut theorems and their applications. SIAM J Comput 25:235–251
Garg N, Vazirani V, Yannakakis M (1997) Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18:3–20
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: 473–496
Keijsper JCM, Pendavingh RA, Stougie L (2006) A linear programming formulation of Mader’s edge- disjoint paths problem. J Comb Theory Ser B 96:159–163
Korte B, Lovász L, Prömel HJ, Schrijver A (eds) (1990) Paths, flows and VLSI-layout. Algorithms and combinatorics 9, Springer, Berlin
Okamura H, Seymour P (1981) Multicommodity flows in planar graphs. J Comb Theory Ser B 31:75–81
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 47:77–80
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bentz, C. Exact and approximate resolution of integral multiflow and multicut problems: algorithms and complexity. 4OR 6, 89–92 (2008). https://doi.org/10.1007/s10288-007-0040-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10288-007-0040-x
Keywords
- Multicuts
- Integral multiflows
- Polynomial-time solvability
- Polynomial approximation
- Combinatorial optimization
- Graph theory