Cyclic group and knapsack facets

Aráoz, Julián; Evans, Lisa; Gomory, Ralph E.; Johnson, Ellis L.

doi:10.1007/s10107-003-0390-x

Cited by 46 publications

(60 citation statements)

References 9 publications

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“…Comparing columns 6 and 7, we see that the two-step mir facets which are not 2slopes are also important facets of P (n, r). Thus, we establish the importance of another class of facets of P (n, r) in addition to the scaled MIRs and 2slopes already discussed in [3,6].…”

Section: Computational Resultssupporting

confidence: 61%

“…We feel that analyzing the polyhedral structure of small master polyhedra can yield families of inequalities which are facet-defining for all master polyhedra, in particular the canonical master polyhedron of Y . This approach has been studied in [3] and [5]. These inequalities could then be used as cutting planes for Y .…”

Section: Discussionmentioning

confidence: 99%

“…A similar study by Evans [6] reports that the so-called "2slope facets" [3] constitute the second most important class of facets . The 2slope facets form a sub-class of the twostep MIR facets (7).…”

Section: Important Facets Of Master Polyhedramentioning

confidence: 97%

“…It is shown in [5] that the "2slope facets" in [3] form a sub-class of scaled two-step MIR facets. A simple transformation can be used to show that Theorem 16 is also true for these 2slope facets, that is, associated template inequalities for Y are dominated by appropriately chosen scaled two-step MIR inequalities.…”

Section: Comparing Template and Mir Inequalitiesmentioning

confidence: 99%

“…Recently, there has been renewed interest in P (n, r), see [14], [3], [13] and [5]. In particular, Evans [6] and Gomory, Johnson and Evans [14] present an empirical approach to identify "important" facets of P (n, r) for small n and propose using sub-additive functions based on these important facets as cutting planes for general integer programs.…”

Section: Introductionmentioning

confidence: 99%

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Valid inequalities based on the interpolation procedure

Dash

Günlük

2005

Math. Program.

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We study the interpolation procedure of , which generates cutting planes for general integer programs from facets of cyclic group polyhedra. This idea has recently been re-considered by Evans (2002) and Gomory, . We compare inequalities generated by this procedure with mixed-integer rounding (MIR) based inequalities discussed in Dash and Gunluk (2003). We first analyze and extend the shooting experiment described in Gomory, Johnson and Evans. We show that MIR based inequalities dominate inequalities generated by the interpolation procedure in some important cases. We also show that the Gomory mixed-integer cut is likely to dominate any inequality generated by the interpolation procedure in a certain probabilistic sense. We also generalize a result of Cornuéjols, Li and Vandenbussche (2003) on comparing the strength of the Gomory mixed-integer cut with related inequalities.

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Section: Computational Resultssupporting

confidence: 61%

Section: Discussionmentioning

confidence: 99%

Section: Important Facets Of Master Polyhedramentioning

confidence: 97%

Section: Comparing Template and Mir Inequalitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Valid inequalities based on the interpolation procedure

Dash

Günlük

2005

Math. Program.

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Inequalities from Group Relaxations

Richard

2011

Wiley Encyclopedia of Operations Research and Management Science

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This article provides a review of the group‐theoretic approach to the generation of cutting planes in mixed integer programming. After motivating the notion of corner relaxation graphically, we give a definition of master group problems. We then present a hierarchy of valid inequalities for master group problems that can be used as cutting planes for mixed integer programs. We describe next various procedures that can be used to obtain the “strongest” valid inequalities for master group problems. We conclude by commenting on the computational possibilities of group‐theoretic cuts in mixed integer programming.

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New inequalities for finite and infinite group problems from approximate lifting

Miller

Richard

2008

Naval Research Logistics

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In this paper, we derive new families of piecewise linear facet-defining inequalities for the finite group problem and extreme inequalities for the infinite group problem using approximate lifting. The new valid inequalities for the finite group problem are two-and three-slope facet-defining inequalities as well as the first family of four-slope facet-defining inequalities. The new valid inequalities for the infinite group problem are families of two-and three-slope extreme inequalities, including nontrivial inequalities that are not continuous. These new inequalities not only illustrate the diversity of strong inequalities for the finite and infinite group problems, but also provide a large variety of new cutting planes for solving integer and mixed-integer programming problems.

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Cyclic group and knapsack facets

Cited by 46 publications

References 9 publications

Valid inequalities based on the interpolation procedure

Valid inequalities based on the interpolation procedure

Inequalities from Group Relaxations

New inequalities for finite and infinite group problems from approximate lifting

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