The maximal dimension of a commutative subalgebra of the Grassmann algebra is determined. It is shown that for any commutative subalgebra there exists an equidimensional commutative subalgebra spanned by monomials. It follows that the maximal dimension of a commutative subalgebra can be expressed in terms of the maximal size of an intersecting system of subsets of odd size in a finite set.2010 MSC: 15A75 (Primary); 05D05 (Secondary); 16W55 (Secondary); 13A02 (Secondary)

Abstract. With the advent of new network architectures, like Software Defined Networks, the rules governing the way traffic is routed through the network are becoming increasingly complex. In this paper, we revisit the theoretic underpinnings of policy routing in the light of the new requirements. We show that certain simple but plausible algebraic properties already induce intractable path selection instances, and we extend the algebraic description of policies for which the related path selection problem is guaranteed to be tractable with a new class, called polynomial finite algebras, which captures many real-life application domains.

Future online services will be provisioned over the federation of infrastructure providers for economic reasons: for effective usage of resources and for a wide geographic reach of customers. The technical enablers of this setup, envisioned also for 5G services, are the virtualization techniques applied in data centers, and in access and core networks. Business aspects around the provisioning of these services however pose still unresolved questions. In this heterogeneous setup those who want to deploy an online service face the problem of selecting the compute and network resource set that fulfills the technical requirements of the service deployment and that is preferable also from an economic point of view. Infrastructure providers compete among themselves for these customers, and shape their business offerings with profit maximization in mind. We model this resource market with the tool set of graph and game theory in order to study its characteristics. We show that customers need to tackle a hard problem if they want to meet both their technical and business needs, let alone minimizing costs. Furthermore, we derive the best pricing strategies that the providers should follow given their expectation about customers’ demand. Interestingly, we show that equilibrium prices and the attainable income strongly depend on the provider’s location within the network.

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