52B12 Special polytopes (linear programming, centrally symmetric, etc.)
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- integer programming (3)
- symmetry breaking (3)
- face numbers of polytopes (2)
- orbitopes (2)
- Delaunay triangulations (1)
- graph coloring (1)
- graph ismorphism (1)
- lexicographic representatives (1)
- lower bound theorem (1)
- perfect graphs (1)
We analyze a remarkable class of centrally symmetric polytopes, the Hansen
polytopes of split graphs. We confirm Kalai's 3^d-conjecture for such polytopes
(they all have at least 3^d nonempty faces) and show that the Hanner polytopes
among them (which have exactly 3^d nonempty faces) correspond to threshold
graphs. Our study produces a new family of Hansen polytopes that have only
3^d+16 nonempty faces.
Orbitopes can be used to handle symmetries which arise in integer programming formulations with an inherent assignment
structure.
We investigate the detection of symmetries appearing in this approach.
We show that detecting so-called orbitopal symmetries is graph-isomorphism hard in general, but can be performed in linear
time if the assignment structure is known.
Orbitopal Fixing
(2007)
The topic of this paper are integer programming models in which a subset of 0/1-variables encode a partitioning of a set of objects into disjoint subsets. Such models can be surprisingly hard to solve by branch-and-cut algorithms if the order of the subsets of the partition is irrelevant. This kind of symmetry unnecessarily blows up the branch-and-cut tree.
We present a general tool, called orbitopal fixing, for enhancing the capabilities of branch-and-cut algorithms in solving such symmetric integer programming models. We devise a linear time algorithm that,
applied at each node of the branch-and-cut tree, removes redundant parts
of the tree produced by the above mentioned symmetry. The method relies on certain polyhedra, called orbitopes, which have been investigated in (Kaibel and Pfetsch 2007). It does, however, not add inequalities to the model, and thus, it does not increase the difficulty of solving the linear programming relaxations. We demonstrate the computational power of orbitopal fixing at the example of a graph partitioning problem motivated from frequency planning in mobile telecommunication networks.
We introduce orbitopes as the convex hulls of 0/1-matrices that are lexicographically maximal subject to a group acting on the
columns. Special cases are packing and partitioning orbitopes, which
arise from restrictions to matrices with at most or exactly one 1-entry
in each row, respectively. The goal of investigating these polytopes is to
gain insight into ways of breaking certain symmetries in integer programs
by adding constraints, e.g., for a well-known formulation of the graph
coloring problem.
We provide a thorough polyhedral investigation of packing and partitioning orbitopes for the cases in which the group acting on the columns
is the cyclic group or the symmetric group. Our main results are complete linear inequality descriptions of these polytopes by facet-defining
inequalities. For the cyclic group case, the descriptions turn out to be
totally unimodular, while for the symmetric group case both the description and the proof are more involved. Nevertheless, the associated
separation problem can be solved in linear time also in this case.