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| Paper IPM / M / 17633 |
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| Abstract: | |
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Cycle polytopes of matroids have been introduced in combinatorial optimization as
a generalization of important classes of polyhedral objects like cut polytopes and
Eulerian subgraph polytopes associated to graphs. Here we start an algebraic and
geometric investigation of these polytopes by studying their toric algebras, called
cycle algebras, and their defining ideals. Several matroid operations are considered
which determine faces of cycle polytopes that belong again to this class of polyhedral
objects. As a key technique used in this paper, we study certain minors of given
matroids which yield algebra retracts on the level of cycle algebras. In particular,
that allows us to use a powerful algebraic machinery. As an application, we study
highest possible degrees in minimal homogeneous systems of generators of defining
ideals of cycle algebras as well as interesting cases of cut polytopes and Eulerian
subgraph polytopes.
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