859 research outputs found
Orbitopal Fixing
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, since this kind of
symmetry unnecessarily blows up the search 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 search tree, removes
redundant parts of the tree produced by the above mentioned symmetry. The
method relies on certain polyhedra, called orbitopes, which have been
introduced bei Kaibel and Pfetsch (Math. Programm. A, 114 (2008), 1-36). It
does, however, not explicitly add inequalities to the model. Instead, it uses
certain fixing rules for variables. We demonstrate the computational power of
orbitopal fixing at the example of a graph partitioning problem.Comment: 22 pages, revised and extended version of a previous version that has
appeared under the same title in Proc. IPCO 200
Structure of the Loday-Ronco Hopf algebra of trees
Loday and Ronco defined an interesting Hopf algebra structure on the linear
span of the set of planar binary trees. They showed that the inclusion of the
Hopf algebra of non-commutative symmetric functions in the Malvenuto-Reutenauer
Hopf algebra of permutations factors through their Hopf algebra of trees, and
these maps correspond to natural maps from the weak order on the symmetric
group to the Tamari order on planar binary trees to the boolean algebra.
We further study the structure of this Hopf algebra of trees using a new
basis for it. We describe the product, coproduct, and antipode in terms of this
basis and use these results to elucidate its Hopf-algebraic structure. We also
obtain a transparent proof of its isomorphism with the non-commutative
Connes-Kreimer Hopf algebra of Foissy, and show that this algebra is related to
non-commutative symmetric functions as the (commutative) Connes-Kreimer Hopf
algebra is related to symmetric functions.Comment: 32 pages, many .eps pictures in color. Minor revision
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