Axiomatizations of quasi-polynomial functions on bounded chains

Two emergent properties in aggregation theory are investigated, namely horizontal maxitivity and comonotonic maxitivity (as well as their dual counterparts) which are commonly defined by means of certain functional equations. We completely describe the function classes axiomatized by each of these properties, up to weak versions of monotonicity in the cases of horizontal maxitivity and minitivity. While studying the classes axiomatized by combinations of these properties, we introduce the concept of quasi-polynomial function which appears as a natural extension of the well-established notion of polynomial function. We give further axiomatizations for this class both in terms of functional equations and natural relaxations of homogeneity and median decomposability. As noteworthy particular cases, we investigate those subclasses of quasi-term functions and quasi-weighted maximum and minimum functions, and provide characterizations accordingly

Online Admission Control and Embedding of Service Chains

The virtualization and softwarization of modern computer networks enables the definition and fast deployment of novel network services called service chains: sequences of virtualized network functions (e.g., firewalls, caches, traffic optimizers) through which traffic is routed between source and destination. This paper attends to the problem of admitting and embedding a maximum number of service chains, i.e., a maximum number of source-destination pairs which are routed via a sequence of to-be-allocated, capacitated network functions. We consider an Online variant of this maximum Service Chain Embedding Problem, short OSCEP, where requests arrive over time, in a worst-case manner. Our main contribution is a deterministic O(log L)-competitive online algorithm, under the assumption that capacities are at least logarithmic in L. We show that this is asymptotically optimal within the class of deterministic and randomized online algorithms. We also explore lower bounds for offline approximation algorithms, and prove that the offline problem is APX-hard for unit capacities and small L > 2, and even Poly-APX-hard in general, when there is no bound on L. These approximation lower bounds may be of independent interest, as they also extend to other problems such as Virtual Circuit Routing. Finally, we present an exact algorithm based on 0-1 programming, implying that the general offline SCEP is in NP and by the above hardness results it is NP-complete for constant L.Comment: early version of SIROCCO 2015 pape

On semiring complexity of Schur polynomials

Semiring complexity is the version of arithmetic circuit complexity that allows only two operations: addition and multiplication. We show that semiring complexity of a Schur polynomial {s_\lambda(x_1,\dots,x_k)} labeled by a partition {\lambda=(\lambda_1\ge\lambda_2\ge\cdots)} is bounded by {O(\log(\lambda_1))} provided the number of variables $k$ is fixed

Pseudo-polynomial functions over finite distributive lattices

In this paper we consider an aggregation model f: X1 x ... x Xn --> Y for arbitrary sets X1, ..., Xn and a finite distributive lattice Y, factorizable as f(x1, ..., xn) = p(u1(x1), ..., un(xn)), where p is an n-variable lattice polynomial function over Y, and each uk is a map from Xk to Y. The resulting functions are referred to as pseudo-polynomial functions. We present an axiomatization for this class of pseudo-polynomial functions which differs from the previous ones both in flavour and nature, and develop general tools which are then used to obtain all possible such factorizations of a given pseudo-polynomial function.Comment: 16 pages, 2 figure