Facets of the linear ordering polytope

Grötschel, Martin; Jünger, Michael; Reinelt, Gerhard

doi:10.1007/bf01582010

Cited by 152 publications

(101 citation statements)

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“…Since the linear ordering variables express a total ordering of the jobs through the constraints (3)-(4), in any feasible solution they satisfy the 3-dicycle inequalities of Grötschel et al (1985) (adapted to our convention that ord jk is defined only if j < k): ord j1,j2 + ord j2,j3 − ord j1,j3 ≤ 1, −ord j1,j2 − ord j2,j3 + ord j1,j3 ≤ 0 j 1 , j 2 , j 3 ∈ J , j 1 < j 2 < j 3 (C1)…”

Section: Cutting Planesmentioning

confidence: 99%

Minimizing the maximum lateness on a single machine with raw material constraints by branch-and-cut

Györgyi

Kis

2018

Computers & Industrial Engineering

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Machine scheduling with raw material constraints has a great practical potential, as it is solved by ad-hoc methods in practice in several manufacturing and logistic environments. In this paper we propose an exact method for solving this problem with the maximum lateness objective based on mathematical programming, our main contribution being a set of new cutting planes that can be used to accelerate a MIP solver. We report on computational results on a wide set of instances.

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Section: Cutting Planesmentioning

confidence: 99%

Minimizing the maximum lateness on a single machine with raw material constraints by branch-and-cut

Györgyi

Kis

2018

Computers & Industrial Engineering

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“…The three-dicycle inequalities define facets of P n LO and completely describe it (together with the trivial inequalities) up to n = 5. For n ≥ 6, many more valid and facet-inducing inequalities are known for P n LO that was intensively studied, e.g, in [42,43,44]. However, for many of them the associated separation problem is itself N Phard [31,44] and sometimes even no practical separation algorithm is known at all.…”

Section: The Linear Ordering Polytopementioning

confidence: 99%

“…For n ≥ 6, the k-fence inequalities define facets of P n LO [43]. They are based on particular orientations of a complete bipartite graph K k,k .…”

Section: The Linear Ordering Polytopementioning

confidence: 99%

“…This is a potential disadvantage since it is likely that these would help in detecting infeasibility of some schedule lengths more quickly, especially because the (feasibility) problem to be solved is indeed to prove that a particular polyhedron contains no integer point. Even more, some of the more complex classes of facet defining inequalities of the linear ordering polytope, such as, e.g., the Möbius ladders [43], are in fact special cases of {0, 1 2 }-cuts [44] that could possibly be recognized this way. Despite knowing about these disadvantages, ABACUS was preferred over a commercial IP optimization software because we did not aim at competing with the CP solver by means of black-box tools.…”

Section: Implementation With Abacusmentioning

confidence: 99%

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An integer programming approach to optimal basic block instruction scheduling for single-issue processors

Jünger

Mallach

2016

Discrete Optimization

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We present a novel integer programming formulation for basic block instruction scheduling on single-issue processors. The problem can be considered as a very general sequential task scheduling problem with delayed precedence-constraints. Our model is based on the linear ordering problem and has, in contrast to the last IP model proposed, numbers of variables and constraints that are strongly polynomial in the instance size. Combined with improved preprocessing techniques and given a time limit of ten minutes of CPU and system time, our branch-and-cut implementation is capable to solve all but eleven of the 369, 861 basic blocks of the SPEC 2000 integer and floating point benchmarks to proven optimality. This is competitive to the current state-of-the art constraint programming approach that has also been evaluated on this test suite.

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“…Pour les méthodes exactes, on peut signaler l'approche polyédrale de M. Grotschel, M. Jünger et G. Reinelt ([36], [37] et [38]) complétée, lorsqu'elle échoue, par une méthode arborescente ; elle a permis à ces auteurs de résoudre de manière exacte des problèmes assez gros, allant, pour des problèmes associés à des données réelles, jusqu'à n = 60.…”

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