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.

Infrared spectra of C2H2 under jet-cooled and para-H2 matrix conditions

In spectra of jet-cooled C2H2 recorded with an FTIR spectrometer, the ν5, ν4 + ν5, ν3 and ν2 + ν4 + ν5 bands all exhibit an intensity distribution corresponding to ∼6 K for rotation, with no evidence of nuclear spin conversion. Spectra of C2H2 isolated in solid p-H2 show no evidence of rotation of C2H2. The strong interaction between ν3 and ν2 + ν4 + ν5 in the gas phase is diminished in solid p-H2. Lines associated with dimer, trimer and tetramer of C2H2 are identified. Spectral features characteristic of solid state acetylene are observed under jet-cooled conditions. © 2007 Elsevier B.V. All rights reserved.

Properties of the protein matrix revealed by the free energy of cavity formation.

The classical picture of the hydrophobic stabilization of proteins invokes a resemblance between the protein interior and nonpolar solvents, but the extent to which this is the case has often been questioned. The protein interior is believed to be at least as tightly packed as organic crystals, and was shown to have very low compressibility. There is also evidence that these properties are not uniform throughout the protein, and conflicting views exist on the nature of sidechain packing and on its influence on the properties of the protein.

Optical spectroscopyof matrix-isolated halogenated fullerenes

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