The hypermetric cone is polyhedral

Deza, Michel; Grishukhin, Viatcheslav; Laurent, Monique

doi:10.1007/bf01303512

Cited by 30 publications

(21 citation statements)

References 16 publications

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“…Deza et al [DGL93] show that this set is in fact polyhedral, but it is currently open whether the separation problem X ∈ HY P can be decided efficiently. The inequalities defining HYP are called hypermetric inequalities.…”

Section: Generic Cutting Planesmentioning

confidence: 99%

Semidefinite Relaxations for Integer Programming

Rendl

2009

50 Years of Integer Programming 1958-2008

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We survey some recent developments in the area of semidefinite optimization applied to integer programming. After recalling some generic modeling techniques to obtain semidefinite relaxations for NP-hard problems, we look at the theoretical power of semidefinite optimization in the context of the Max-Cut and the Coloring Problem. In the second part, we consider algorithmic questions related to semidefinite optimization, and point to some recent ideas to handle large scale problems. The survey is concluded with some more advanced modeling techniques, based on matrix relaxations leading to copositive matrices.

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Section: Generic Cutting Planesmentioning

confidence: 99%

Semidefinite Relaxations for Integer Programming

Rendl

2009

50 Years of Integer Programming 1958-2008

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“…Another important open question is whether the gap inequalities define a polyhedron. (It is known that the hypermetric and rounded psd inequalities define polyhedra [8,20], whereas the negative type and psd inequalities do not [9,18].) Finally, it would of course be nice to have an explicit example of a gap inequality with ( ) > 1 that is not implied by rounded psd inequalities.…”

Section: On the Separation Problemmentioning

confidence: 99%

Complexity results for the gap inequalities for the max-cut problem

Galli

Kaparis

Letchford

2012

Operations Research Letters

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In 1996, Laurent and Poljak introduced an extremely general class of cutting planes for the max-cut problem, called gap inequalities. We prove several results about them, including the following: (i) there must exist non-dominated gap inequalities with gap larger than 1, unless = co-; (ii) there must exist non-dominated gap inequalities with exponentially large coefficients, unless = co-; (iii) the separation problem for gap inequalities can be solved in finite time (specifically, doubly exponential time).

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“…In fact, QHY P n is polyhedral (see [DGL93]) and the triangle inequality is redundant. The smallest case when QHY P n is a proper sub-cone of QMET n is n = 5; see some information on QHY P 5 in Table 1 and http://www.geomappl.ens.fr/~dutour/ 6 Appendix 1: Extreme rays of the cone QM ET 5 By direct computation 43590 extreme rays of QMET 5 were found.…”

Section: The Case Of Pointsmentioning

confidence: 99%