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On the set covering polyhedron of circulant matrices. (English) Zbl 1203.90127
Summary: A well known family of minimally nonideal matrices is the family of the incidence matrices of chordless odd cycles. A natural generalization of these matrices is given by the family of circulant matrices. Ideal and minimally nonideal circulant matrices have been completely identified by G. Cornuéjols and B. Novick [J. Comb. Theory, Ser. B 60, No. 1, 145–147 (1994; Zbl 0794.05077)]. In this work we classify circulant matrices and their blockers in terms of the inequalities involved in their set covering polyhedra. We exploit the results due to Cornuéjols and Novick in the above-cited reference for describing the set covering polyhedron of blockers of circulant matrices. Finally, we point out that the results found on circulant matrices and their blockers present a remarkable analogy with a similar analysis of webs and antiwebs due to A. Pêcher and A. K. Wagler [Eur. J. Comb. 27, No. 7, 1172–1185 (2006; Zbl 1102.90053); Math. Program. 105, 311–328 (2006; Zbl 1083.05032)] and A. K. Wagler [in: The sharpest cut. Impact of Manfred Padberg and his work, in: MPS/SIAM Series on Optimization 4, 77–96 (2004; Zbl 1072.05026); 4OR 2, No. 2, 149–152 (2004; Zbl 1112.05082)].
MSC:
90C27Combinatorial optimization
52B12Special polytopes (linear programming, centrally symmetric, etc.)
15B57Hermitian, skew-Hermitian, and related matrices
05B20Matrices (incidence, Hadamard, etc.)
05C50Graphs and linear algebra
05C17Perfect graphs
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