Abstract
We give polynomial algorithms for the fractional covering problems for forests andb-matchings: min{1·y: yA≥w,y≥0} whereA is a matrix whose rows are the incidence vectors of forests/b-matchings respectively. It is shown that each problem can be solved by a series of max-flow/min-cut calculations, and hence the use of the ellipsoid algorithm to guarantee a polynomial algorithm can be avoided.
Similar content being viewed by others
References
C. Berge,Graphs and hypergraphs (North-Holland, Amsterdam, 1973).
W.H. Cunningham, “Testing membership in matroid polyhedra”. Technical Report 81207-OR, Institut für Operations Research, Universitát Bonn (1981); to appear inJournal of Combinatorial Theory B.
J. Edmonds, “Maximum matching and a polyhedron with 0–1 vertices”,Journal of Research. National Bureau of Standards 69B (1975) 125–130.
J. Edmonds, “Submodular functions, matroids and certain polyhedra”, in: R. Guy, H. Hanani, N. Sauer and J. Schonheim, eds.Combinatorial structures and their applications. (Gordon and Breach, New York, 1969).
D.R. Fulkerson, “Blcoking and antiblocking pairs of polyhedra”,Mathematical Programming 1 (1971) 168–194.
P. Gacs and L. Lovász, “Khachian's algorithm for linear programming”,Mathematical Programming Study 14 (1981) 61–68.
M. Grötschel, L. Lovász and A. Schrijver, “The ellipsoid method and its consequences in combinatorial optimization”,Combinatorica 1 (1981) 169–197.
P. Hansen, “Method of nonlinear 0–1 programming”,Annals of Discrete Mathematics 5 (1979) 53–70.
L.G. Khachian, “A polynomial algorithm in linear programming”,Doklady Akademii Nauk USSR 244 (1979). Translated inSoviet Mathematics Doklady 20 (1979) 191–194.
M.W. Padberg and M.R. Rao, “The Russian method for linear inequalities III: Bounded integer programming”, GBA Working Paper, New York University, May, 1981.
M.W. Padberg and L.A. Wolsey, “Trees and cuts”. CORE Discussion Paper 8138. October, 1981.
M.W. Padberg and M.R. Rao, “Odd minimum cut-sets and b-matching”,Mathematics of Operations Research 7 (1982) 67–80.
J.C. Picard and H.D. Ratcliff, “Minimum cuts and related problems”,Networks 5 (1975) 357–370.
J.C. Picard and M. Queyranne, “Selected applications of minimum cuts in networks”,Infor 20 (1982) 394–422.
D.B. Weinberger, “Network flows minimum coverings and the four-color conjecture”,Operations Research 24 (1976) 272–290.
Author information
Authors and Affiliations
Additional information
Visiting professor at the European Institute for Advanced Studies in Management in Brussels and at CORE. Supported in part by the CIM. On leave from New York University, New York, NY 10006.
Rights and permissions
About this article
Cite this article
Padberg, M.W., Wolsey, L.A. Fractional covers for forests and matchings. Mathematical Programming 29, 1–14 (1984). https://doi.org/10.1007/BF02591725
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02591725