A combinatorial study of partial order polytopes
European Journal of Combinatorics
Facets of linear signed order polytopes
Discrete Applied Mathematics
Lagrangian heuristics for the linear ordering problem
Metaheuristics
Revised GRASP with path-relinking for the linear ordering problem
Journal of Combinatorial Optimization
Discrete Optimization
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The linear ordering problem has many applications and was studied by many authors (for a survey, see [P. Fishburn, SIAM J. Discrete Math., 4 (1990), pp. 478--488; M. Grötschel, M. Jünger, and G. Reinelt, Math. Programming, 33 (1985), pp. 43--60; G. Reinelt, The Linear Ordering Problem: Algorithm and Applications, Heldermann Verlag, 1985]). One approach to solving this problem, the so-called cutting plane method [M. Grötschel, M. Jünger, and G. Reinelt, Oper. Res., 32 (1984), pp. 1195--1220; V. A. Yemelichev, M. M. Kovalev, and M. K. Kravtsov, Polytopes, Graphs, Optimization, Cambridge University Press, 1984], derives facet-defining inequalities which are violated by current nonfeasible solution and adds them to the system of inequalities of current linear programming problems.We present a method (rotation method) for generating new facets of polyhedra by using previously known ones. The rotation method for the linear ordering polytope generalizes facets induced by subgraphs called $m$-fences, Möbius ladders, and Zm-facets introduced by Reinelt, (m,k)-fences introduced by Bolotashvili [G. C. Bolotashvili, A Class of Facets of the Permutation Polytope and a Method for Constructing Facets of the Permutation Polytope, preprint VINITI N3403-B87, Moscow, 1987 (in Russian)]; and t-reinforced m-fences introduced by Leung and Lee [J. Leung and J. Lee, Discrete Appl. Math., 50 (1984), pp. 185--200]. We introduce 10 collections of inequalities representing facets of the linear ordering polytope. Among them are three that coincide with earlier known ones: m-wheel-facets introduced by Reinelt, augmented m-fences introduced by McLennan [A. McLennan, in Preferences, Uncertainty and Optimality, West View Press, 1990, pp. 187--202]; and augmented t-reinforced m-fences introduced by Leung and Lee.