Abstract
It is shown in this paper how a solution for a combinatorial problem obtained from applying the greedy algorithm is guaranteed to be optimal for those instances of the problem that, under an appropriate algebraic representation, satisfy the Cohen-Macaulay property known for rings and modules in Commutative Algebra. The choice of representation for the instances of a given combinatorial problem is fundamental for recognizing the Cohen-Macaulay property. Departing from an exposition of the general framework of simplicial complexes and their associated Stanley-Reisner ideals, wherein the Cohen-Macaulay property is formally defined, a review of other equivalent frameworks more suitable for graphs or arithmetical problems will follow. In the case of graph problems a better framework to use is the edge ideal of Rafael Villarreal. For arithmetic problems it is appropriate to work within the semigroup viewpoint of toric geometry developed by Antonio Campillo and collaborators.
For Antonio Campillo and Miguel Angel Revilla
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Notes
- 1.
The reader is encouraged to prove this equivalence.
- 2.
The dimension of a finitely generated ring is the maximum cardinality of an algebraically independent set. This is equivalent in R Δ to dim(Δ) + 1 and Eq. (2).
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Acknowledgements
The author acknowledges the support of the Ministerio de Economía, Industria y Competitividad, Spain, project MACDA [TIN2017-89244-R] and of the Generalitat de Catalunya, project MACDA [SGR2014-890]
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Arratia, A. (2018). The Greedy Algorithm and the Cohen-Macaulay Property of Rings, Graphs and Toric Projective Curves. In: Greuel, GM., Narváez Macarro, L., Xambó-Descamps, S. (eds) Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics. Springer, Cham. https://doi.org/10.1007/978-3-319-96827-8_17
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