Abstract
Graphs for which the set oft-joins andt-cuts has “the max-flow-min-cut property”, i.e. for which the minimal defining system of thet-join polyhedron is totally dual integral, have been characterized by Seymour. An extension of this problem isto characterize the (uniquely existing) minimal totally dual integral defining system (Schrijver-system) of an arbitrary t-join polyhedron. This problem is solved in the present paper. The main idea is to uset-borders, a generalization oft-cuts, to obtain an integer minimax formula for the cardinality of a minimumt-join. (At-border is the set of edges joining different classes of a partition of the vertex set intot-odd sets.) It turns out that the (uniquely existing) “strongest minimax theorem” involves just this notion.
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References
W. Cook, A minimal totally dual integral defining system for theb-matching polyhedron,SIAM Journal on Alg. and Disc. Methods,4 (1983), 212–220.
W. Cook, On Some Aspects of Totally Dual Integral Systems,Ph. D. Thesis, University of Waterloo, Canada, 1983.
W. Cook, A note on matchings and separability,Report 84315-OR, Institut für Ökonometrie und Operations Research, Bonn, W. Germany, 1984.
W. Cook andW. R. Pulleyblank, Linear Systems for Constrained Matching Problems,Report No. 84323-OR, Institut für Ökonometrie und Operations Research, Bonn, W. Germany, 1984.
W. H. Cunningham andA. B. Marsh III, A primal algorithm for optimum matching,Math. Prog. Study. 8 (1978), 50–72.
J. Edmonds andE. L. Johnson, Matching: a well-solved class of integer linear programs, in:R. Guy, H. Hanani, N. Sauer, andJ. Schonheim (eds..Combinatorial Structures and Their Applications (Gordon and Breach, New York, 1970), 89–92.
J. Edmonds andE. L. Johnson, Matching, Euler tours and the Chinese postman,Math Programming,5 (1973), 88–124.
J. Edmonds, L. Lovász andW. R. Pulleyblank, Brick Decompositions and the Matching Rank of Graphs,Combinatorica,2 (1982), 247–274.
A. Frank, A. Sebő andÉ. Tardos, Covering Directed and Odd Cuts,Mathematical Programming Study.22 (1984), 99–112.
A.Gerards, A.Schrijver, Signed Graphs—Regular Matroids—Grafts,Tilburg University, Department of Econometrics. Tilburg.
A.Gerards, A.Sebő, Total dual integrality implies local strong unimodularity,Mathematical Programming.
A.Gerards,private communication.
M. Grötschel, L. Lovász andA. Schrijver, The ellipsoid method and its consequences in combinatorial optimization,Combinatorica.1 (1981), 169–197.
E.Korach, Packing ofT-cuts and Other Aspects of Dual Integrality,Ph. D. Thesis, Waterloo University. 1982.
L. Lovász, On the structure of factorizable graphs,Acta Math. Acad. Sci. Hung.,23 (1972), 179–195.
L. Lovász, 2-matchings and 2-covers of hypergraphs,Acta Mathematica Academiae Scientiarum Hungaricae.26 (1975), 433–444.
L.Lovász and M. D.Plummer, On bicritical graphs, Infinite and Finite Sets,Colloqu. Math. Soc. J. Bolyai 10,Budapest (A. Hajnal, R. Rado and V. T. Sós et al (eds.)), (1975), 1051–1079.
L.Lovász and M. D.Plummer,Matching Theory, Akadémiai Kiadó, 1985.
Mei Gu Guan, Graphic programming using odd or even points,Chinese Math.,1 (1962), 273–277.
W. R. Pulleyblank, Dual integrality inb-matching problems,Math. Prog. Study,12 (1980), 176–196.
W. R. Pulleyblank, Total dual integrality andb-matchings,Operations Research Letters,1 (1981), 28–30.
A. Schrijver, On Cutting Planes,Annals Discrete Math.,9 (1980), 291–296.
A. Schrijver, On total dual integrality,Lin. Alg. and its Appl.,38 (1981), 27–32.
A. Schrijver, Min-max results in combinatorial optimization, in:A. Bachern, M. Grötschel, andB. Karte (eds..Mathematical Programming —The State of the Art (Springer Verlag, Heidelberg, 1983), 439–500.
A. Schrijver,Theory of linear and integer programming, Wiley, Chichester, (1986).
A.Sebő, Undirected Distances and the Postman Structure of Graphs,to appear in Journal of Combinatorial Theory, 1988.
A. Sebő, Finding thet-join Structure of Graphs,Mathematical Programming,36 (1986), 123–134.
A.Sebő, The Chinese Postman Problem: Algorithms, Structure and Applications,SZTAKI Working Paper, MO/61, (1985).
A. Sebő, A quick proof of Seymour's theorem ont-joins,Discrete Mathematics,64 (1987), 101–103.
A.Sebő, C. Sc. thesis(in Hungarian), august 1987.
P. D. Seymour, The matroids with the Max-Flow-Min-Cut Property,J. Comb. Theory, Series B.23 (1977), 189–222.
P. D. Seymour, On odd cuts and plane multicommodity flows,Proc. London Math. Soc., 3/42 (1981), 178–192.
É.Tardos,private communication.
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On leave from Eötvös Loránd University and Computer and Automation Institute, Budapest.
Supported by Sonderforschungsbereich 303 (DFG), Institut für Operations Research, Universität Bonn, W. Germany