Extended formulations, nonnegative factorizations, and randomized communication protocols

An extended formulation of a polyhedron P is a linear description of a polyhedron Q together with a linear map π such that π(Q)=P. These objects are of fundamental importance in polyhedral combinatorics and optimization theory, and the subject of a number of studies. Yannakakis’ factorization theorem (Yannakakis in J Comput Syst Sci 43(3):441–466, 1991) provides a surprising connection between extended formulations and communication complexity, showing that the smallest size of an extended formulation of $$P$$P equals the nonnegative rank of its slack matrix S. Moreover, Yannakakis also shows that the nonnegative rank of S is at most 2^c, where c is the complexity of any deterministic protocol computing S. In this paper, we show that the latter result can be strengthened when we allow protocols to be randomized. In particular, we prove that the base-2 logarithm of the nonnegative rank of any nonnegative matrix equals the minimum complexity of a randomized communication protocol computing the matrix in expectation. Using Yannakakis’ factorization theorem, this implies that the base-2 logarithm of the smallest size of an extended formulation of a polytope P equals the minimum complexity of a randomized communication protocol computing the slack matrix of P in expectation. We show that allowing randomization in the protocol can be crucial for obtaining small extended formulations. Specifically, we prove that for the spanning tree and perfect matching polytopes, small variance in the protocol forces large size in the extended formulation.

A Multi-Objective Approach for VNE: Problems using multiple ILP formulations

Network Virtualization is a key technology for the Future Internet, allowing the deployment of multiple independent virtual networks that use resources of the same basic infrastructure. An important challenge in the dynamic provision of virtual networks resides in the optimal allocation of physical resources (nodes and links) to requirements of virtual networks. This problem is known as Virtual Network Embedding (VNE). For the resolution of this problem, previous research has focused on designing algorithms based on the optimization of a single objective. On the contrary, in this work we present a multi-objective algorithm, called VNE-MO-ILP, for solving dynamic VNE problem, which calculates an approximation of the Pareto Front considering simultaneously resource utilization and load balancing. Experimental results show evidences that the proposed algorithm is better or at least comparable to a state-of-the-art algorithm. Two performance metrics were simultaneously evaluated: (i) Virtual Network Request Acceptance Ratio and (ii) Revenue/Cost Relation. The size of test networks used in the experiments shows that the proposed algorithm scales well in execution times, for networks of 84 nodes

CIL 2.5041 – Decree of L. Aemilius Paulus - Bronze tablet

Photograph

CIL 5.7749 – Sententia Minuciorum on a dispute over public land between the Genovana and the Langensian Viturians - Bronze tablet

Photograph + clipping from Oggi, May 10, 1952

CIL 6.40890/ IG 14.951 – Bronze tablet with a bilingual ‘senatus consultum’ granting a tax exemption and other privileges to Asklepiades of Klazomenai and other Greek naval captains

Photographs

CIL 10.104 – Senatus consultum de Bacchanalibus, bronze tablet

Photograph

CIL 11.1146 – Lex Rubria de Gallia Cisalpina, bronze tablet, ca. 49 BC

Photographs and notes by A.E. Gordon

CIL 2.1963 – Lex Salpensana, bronze tablet with a municipal law concerning citizenship and manumission

Photograph and translation

CIL 2.1964 – Lex Malacitana, bronze tablet with a municipal law

Photograph and translation

Analysis of β-tricalcium phosphate granules prepared with different formulations by nano-computed tomography and scanning electron microscopy

International audience

Voltammetric determination of L-dopa using an electrode modified with trinuclear ruthenium ammine complex (Ru-red) supported on Y-type zeolite

The L-dopa is the immediate precursor of the neurotransmitter dopamine. Unlike dopamine, L-dopa easily enters the central nervous system and is used in the treatment of Parkinson's disease. A sensitive and selective method is presented for the voltammetric determination of L-dopa in pharmaceutical formulations using a carbon paste electrode modified with trinuclear ruthenium ammine complex [(NH3)(5)Ru-III-O-Ru-IV(NH3)(4)-O-Ru-III(NH3)(5)](6+) (Ru-red) incorporated in NaY zeolite. The parameters which influence on the electrode response (paste composition, potential scan rate, pH and interference) were also investigated. The optimum conditions were found to an electrode composition (m/m) of 25% zeolite containing 6.7% Ru, 50% graphite and 25% mineral oil in acetate buffer at pH 4.8. Voltammetric peak currents showed a linear response for L-dopa concentration in the range between 1.2 x 10(-4) and 1.0 x 10(-2) Mol l(-1) (r = 0.9988) with a detection limit of 8.5 x 10(-5) mol l(-1). The variation coefficient for a 1.0 x 10(-3) mol l(-1) L-dopa (n = 10) was 5.5%. The results obtained for L-dopa in pharmaceutical formulations (tablet) was in agreement with compared official method. In conclusion, this study has illustrated that the proposed electrode modified with Ru-red incorporated zeolite is suitable valuable for selective measurements of L-dopa. (C) 2004 Elsevier B.V. All rights reserved.