Polytopes associated with symmetry handling

Hojny, Christopher; Pfetsch, Marc E.

doi:10.1007/s10107-018-1239-7

“…Therefore, in recent years several new methods for exploiting symmetries in integer linear programming have been developed. See, for example, [29,12,5,22,27,32,14,11,20] and the surveys by Margot [30] and Pfetsch and Rehn [33] for an overview. These methods (with the exception of [11]) fall broadly into two classes: Either they modify the standard branching approach, using isomorphism tests or isomorphism free generation to avoid solving equivalent subproblems, or they use techniques to cut down the original symmetric problem to a less symmetric one, which contains at least one element of each orbit of solutions.…”

Section: A P P L I C At I O N T O I N T E G E R L I N E a R O P T I Mmentioning

confidence: 99%

Equivalence of Lattice Orbit Polytopes

Ladisch¹,

Schürmann²

2018

SIAM J. Appl. Algebra Geometry

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A b s t r ac t . Let G be a finite permutation group acting on R d by permuting coordinates. A core point (for G) is an integral vector z ∈ Z d such that the convex hull of the orbit Gz contains no other integral vectors but those in the orbit Gz. Herr, Rehn and Schürmann considered the question for which groups there are infinitely many core points up to translation equivalence, that is, up to translation by vectors fixed by the group. In the present paper, we propose a coarser equivalence relation for core points called normalizer equivalence. These equivalence classes often contain infinitely many vectors up to translation, for example when the group admits an irrational invariant subspace or an invariant irreducible subspace occurring with multiplicity greater than 1. We also show that the number of core points up to normalizer equivalence is finite if G is a so-called QI-group. These groups include all transitive permutation groups of prime degree. We give an example to show how the concept of normalizer equivalence can be used to simplify integer convex optimization problems. U n i v e r s i tät Ro s t o c k , I n s t i t u t f ü r M at h e m at i k , U l m e n s t r . 6 9 , H au s 3 , 1 8 0 5 7 Ro s t o c k , G e r m a n y

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“…Symmetry-breaking inequalities can be derived from the linear description of the convex hull of an arbitrary representative set [16]. In most works, each chosen representative x is lexicographically maximal in its orbit, i.e., x ľ gpxq, for each g P G. The convex hull of the latter representative set is called the symmetry-breaking polytope [16]. When x is a matrix and when the symmetry group G acts on the columns of x, the symmetry-breaking polytope is called orbitope.…”

Section: Introductionmentioning

confidence: 99%

“…When x is a matrix and when the symmetry group G acts on the columns of x, the symmetry-breaking polytope is called orbitope. Even if complete linear descriptions for orbitopes may be hard to reach in general [26], integer programming formulations for these polytopes still yield full symmetry-breaking inequalities [16]. Instead of considering orbits of solutions, [23,24] introduce inequalities enforcing a lexicographical order within orbits of variables.…”

Section: Introductionmentioning

confidence: 99%

Symmetry-breaking inequalities for ILP with structured sub-symmetry

Bendotti

¹

,

Fouilhoux

²

,

Rottner

³

2020

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We consider integer linear programs whose solutions are binary matrices and whose (sub-)symmetry groups are symmetric groups acting on (sub-)columns. Such structured subsymmetry groups arise in important classes of combinatorial problems, e.g. graph coloring or unit commitment. For a priori known (sub-)symmetries, we propose a framework to build (sub-)symmetry-breaking inequalities for such problems, by introducing one additional variable per considered sub-symmetry group. The derived inequalities are full-symmetry-breaking and in polynomial number w.r.t. the number of sub-symmetry groups considered. The proposed framework is applied to derive such inequalities when the symmetry group is the symmetric group acting on the columns. It is also applied to derive sub-symmetry-breaking inequalities for the graph coloring problem. Experimental results give insight into how to select the right inequality subset in order to efficiently break sub-symmetries. Finally, the framework is applied to derive (sub-)symmetry breaking inequalities for Min-up/min-down Unit Commitment Problem with or without ramp constraints. We show the effectiveness of the approach by presenting an experimental comparison with state-of-the-art symmetry-breaking formulations.

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“…The latter inequalities enforce that the total number of ones in each column is non-increasing, thus not guaranteeing lexicographically non increasing columns for the representatives. An alternative avoiding the exponential coefficients of Friedman's inequalities can be to use the full symmetry-breaking inequalities discussed in [16]. These inequalities ensure that any integer point is in the full orbitope.…”

Section: Introductionmentioning

confidence: 99%

Sub-Symmetry-Breaking Inequalities for ILP with Structured Symmetry

Bendotti

¹

,

Fouilhoux²,

Rottner

³

2019

Integer Programming and Combinatorial Optimization

1

0

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We consider integer linear programs whose solutions are binary matrices and whose (sub-)symmetry groups are symmetric groups acting on (sub-)columns. Such structured subsymmetry groups arise in important classes of combinatorial problems, e.g. graph coloring or unit commitment. For a priori known (sub-)symmetries, we propose a framework to build (sub-)symmetry-breaking inequalities for such problems, by introducing one additional variable per considered sub-symmetry group. The derived inequalities are full-symmetry-breaking and in polynomial number w.r.t. the number of sub-symmetry groups considered. The proposed framework is applied to derive such inequalities when the symmetry group is the symmetric group acting on the columns. It is also applied to derive sub-symmetry-breaking inequalities for the graph coloring problem. Experimental results give insight into how to select the right inequality subset in order to efficiently break sub-symmetries. Finally, the framework is applied to derive (sub-)symmetry breaking inequalities for Min-up/min-down Unit Commitment Problem with or without ramp constraints. We show the effectiveness of the approach by presenting an experimental comparison with state-of-the-art symmetry-breaking formulations.

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Polytopes associated with symmetry handling

Cited by 36 publications

References 38 publications

Equivalence of Lattice Orbit Polytopes

Equivalence of Lattice Orbit Polytopes

Symmetry-breaking inequalities for ILP with structured sub-symmetry

Sub-Symmetry-Breaking Inequalities for ILP with Structured Symmetry

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