New Facets of the Linear Ordering Polytope

Abstract

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. Grotschel, M. Junger, 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. Grotschel, M. Junger, 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, Mobius 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.