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On the 0,1 facets of the set covering polytope. (English) Zbl 0675.90054
Necessary and sufficient conditions for an inequality with 0-1 coefficients to define a facet of the set covering polytope associated with the 0-1 constraint matrix A are given. The work is influenced by the results on the set packing problem, where also the notations are defined in the intersection graph of A. For the case that A is a matrix of odd order and the matrix H<A contains exactly two ones per row and per column it is shown how to decide in polynomial time whether j=1 n x j >(n+1)/2 is valid. This yields a facet of the set covering polytope.
Reviewer: N.Yanev
MSC:
90C09Boolean programming
52BxxPolytopes and polyhedra
05C70Factorization, etc.
90C35Programming involving graphs or networks
90C10Integer programming
References:
[1]E. Balas, ”Set covering with cutting planes from conditional bounds,” in: A. Prekopa, ed.,Survey of Mathematical Programming (North-Holland, Amsterdam, 1979) pp. 393–422.
[2]E. Balas, ”Cutting planes from conditional bounds: A new approach to set covering,”Mathematical Programming Study 12 (1980) 19–36.
[3]E. Balas and A. C. Ho, ”Set covering algorithms using cutting planes, heuristics, and subgradient optimization: a computational study,”Mathematical Programming Study 12 (1980) 37–60.
[4]E. Balas and M. Padberg, ”Set Partitioning: A Survey,”SIAM Review 18 (1976) 710–760. · Zbl 0347.90064 · doi:10.1137/1018115
[5]E. Balas and Shu-Ming Ng, ”Some classes of facets of the set covering polytope,” Research Report, Graduate School of Industrial Administration, Carnegie Mellon University (Pittsburgh, PA, 1984).
[6]E. Balas and Shu-Ming Ng, ”On the set covering polytope: I. All the facets with coefficients in {0, 1, 2},” Research Report MSRR-522, Graduate School of Industrial Administration, Carnegie Mellon University (Pittsburgh, PA, 1986).
[7]E. Balas and E. Zemel, ”Critical cutsets of graphs and canonical facets of set-packing polytopes,”Mathematics of Operations Research 2 (1977) 15–19. · doi:10.1287/moor.2.1.15
[8]C. Berge, ”Balanced Matrices,”Mathematical Programming 2 (1972) 19–31. · Zbl 0247.05126 · doi:10.1007/BF01584535
[9]D.C. Cho, M. Padberg and M.R. Rao, ”On the uncapacitated plant location problem. II. Facets and Lifting theorems,”Mathematics of Operations Research 8 (1983) 590–612. · Zbl 0536.90030 · doi:10.1287/moor.8.4.590
[10]V. Chvàtal, ”On certain polytopes associated with graphs,”Journal of Combinatorial Theory B 18 (1975) 138–154. · Zbl 0277.05139 · doi:10.1016/0095-8956(75)90041-6
[11]M. Conforti, D.G. Corneil and A.R. Mahjoub, ”K i-covers I: Complexity and polytopes,”Discrete Mathematics 58 (1986) 121–142. · Zbl 0584.05052 · doi:10.1016/0012-365X(86)90156-1
[12]G. Cornuéjols and J.M. Thizy, ”Some facets of the simple plant location polytope,”Mathematical Programming 23 (1982) 50–74. · Zbl 0485.90069 · doi:10.1007/BF01583779
[13]D.R. Fulkerson, A.J. Hoffman and R. Oppenheim, ”On balanced matrices,”Mathematical Programming Study 1 (1974) 120–132.
[14]J. Hooker, ”Generalized resolution and cutting planes,” Research Report n. MSRR 524, Graduate School of Industrial Administration, Carnegie Mellon University (Pittsburgh, PA, 1986).
[15]M. Laurent, ”A generalization of antiwebs to independence systems and their relation to set covering polytopes,” Technical Report, CNET PAATIM (Issy les Moulineaux, France, 1987).
[16]G.L. Nemhauser and L.E. Trotter, ”Properties of vertex packing and independence system polyhedra,”Mathematical Programming 6 (1974) 48–61. · Zbl 0281.90072 · doi:10.1007/BF01580222
[17]M.W. Padberg, ”On the facial structure of set packing polyhedra,”Mathematical Programming 5 (1973) 199–215. · Zbl 0272.90041 · doi:10.1007/BF01580121
[18]A. Sassano, ”On the facial structure of the set covering polytope,” IASI-CNR Report nr. 132 (Rome, 1985).
[19]L.E. Trotter, ”A class of facet producing graphs for vertex packing polyhedra,”Discrete Mathematics 12 (1975) 373–388. · Zbl 0314.05111 · doi:10.1016/0012-365X(75)90077-1
[20]L.A. Wolsey, ”Further facet generating procedures for vertex packing polytopes,”Mathematical Programming 11 (1976) 158–163. · Zbl 0348.90148 · doi:10.1007/BF01580383