Tree Projections and Structural Decomposition Methods: Minimality and Game-Theoretic Characterization

Tree projections provide a mathematical framework that encompasses all the various (purely) structural decomposition methods that have been proposed in the literature to single out classes of nearly-acyclic (hyper)graphs, such as the tree decomposition method, which is the most powerful decomposition method on graphs, and the (generalized) hypertree decomposition method, which is its natural counterpart on arbitrary hypergraphs. The paper analyzes this framework, by focusing in particular on "minimal" tree projections, that is, on tree projections without useless redundancies. First, it is shown that minimal tree projections enjoy a number of properties that are usually required for normal form decompositions in various structural decomposition methods. In particular, they enjoy the same kind of connection properties as (minimal) tree decompositions of graphs, with the result being tight in the light of the negative answer that is provided to the open question about whether they enjoy a slightly stronger notion of connection property, defined to speed-up the computation of hypertree decompositions. Second, it is shown that tree projections admit a natural game-theoretic characterization in terms of the Captain and Robber game. In this game, as for the Robber and Cops game characterizing tree decompositions, the existence of winning strategies implies the existence of monotone ones. As a special case, the Captain and Robber game can be used to characterize the generalized hypertree decomposition method, where such a game-theoretic characterization was missing and asked for. Besides their theoretical interest, these results have immediate algorithmic applications both for the general setting and for structural decomposition methods that can be recast in terms of tree projections

Elliptic Flow and Shear Viscosity of the Shattered Color Glass Condensate

In this talk, we report on our results about the computation of the elliptic flow of the quark-gluon-plasma produced in relativistic heavy ion collisions, simulating the expansion of the fireball by solving the relativistic Boltzmann equation for the parton distribution function tuned at a fixed shear viscosity to entropy density ratio $\eta/s$. We emphasize the role of saturation in the initial gluon spectrum modelling the shattering of the color glass condensate, causing the initial distribution to be out of equilibrium. We find that the saturation reduces the efficiency in building-up the elliptic flow, even if the thermalization process is quite fast $\tau_{therm} \approx 0.8 \,\rm fm/c$. and the pressure isotropization even faster $\tau_{isotr} \approx 0.3 \,\rm fm/c$. The impact of the initial non-equilibrium manifests for non-central collisions and can modify the estimate of the viscosity respect to the assumption of full thermalization in $p_T$-space.Comment: 8 pages, 3 figures. Talk given at XIV Convegno su Problemi di Fisica Nucleare Teorica, 29-31 October 2013, Cortona, Ital

Tractable Optimization Problems through Hypergraph-Based Structural Restrictions

Several variants of the Constraint Satisfaction Problem have been proposed and investigated in the literature for modelling those scenarios where solutions are associated with some given costs. Within these frameworks computing an optimal solution is an NP-hard problem in general; yet, when restricted over classes of instances whose constraint interactions can be modelled via (nearly-)acyclic graphs, this problem is known to be solvable in polynomial time. In this paper, larger classes of tractable instances are singled out, by discussing solution approaches based on exploiting hypergraph acyclicity and, more generally, structural decomposition methods, such as (hyper)tree decompositions

Space-charge effects in high-energy photoemission

Pump-and-probe photoelectron spectroscopy (PES) with femtosecond pulsed sources opens new perspectives in the investigation of the ultrafast dynamics of physical and chemical processes at the surfaces and interfaces of solids. Nevertheless, for very intense photon pulses a large number of photoelectrons are simultaneously emitted and their mutual Coulomb repulsion is sufficiently strong to significantly modify their trajectory and kinetic energy. This phenomenon, referred as space-charge effect, determines a broadening and shift in energy for the typical PES structures and a dramatic loss of energy resolution. In this article we examine the effects of space charge in PES with a particular focus on time-resolved hard X-ray (~10 keV) experiments. The trajectory of the electrons photoemitted from pure Cu in a hard X-ray PES experiment has been reproduced through $N$-body simulations and the broadening of the photoemission core-level peaks has been monitored as a function of various parameters (photons per pulse, linear dimension of the photon spot, photon energy). The energy broadening results directly proportional to the number $N$ of electrons emitted per pulse (mainly represented by secondary electrons) and inversely proportional to the linear dimension $a$ of the photon spot on the sample surface, in agreement with the literature data about ultraviolet and soft X-ray experiments. The evolution in time of the energy broadening during the flight of the photoelectrons is also studied. Despite its detrimental consequences on the energy spectra, we found that space charge has negligible effects on the momentum distribution of photoelectrons and a momentum broadening is not expected to affect angle-resolved experiments. Strategy to reduce the energy broadening and the feasibility of hard X-ray PES experiments at the new free-electron laser facilities are discussed.Comment: 15 pages, 2 tables, 8 figure

Thermalization, Isotropization and Elliptic Flow from Nonequilibrium Initial Conditions with a Saturation Scale

In this article we report on our results about the computation of the elliptic flow of the quark-gluon-plasma produced in relativistic heavy ion collisions, simulating the expansion of the fireball by solving the relativistic Boltzmann equation for the parton distribution function tuned at a fixed shear viscosity to entropy density ratio $\eta/s$. Our main goal is to put emphasis on the role of a saturation scale in the initial gluon spectrum, which makes the initial distribution far from a thermalized one. We find that the presence of the saturation scale reduces the efficiency in building-up the elliptic flow, even if the thermalization process is quite fast $\tau_{therm} \approx 0.8 \,\rm fm/c$ and the pressure isotropization even faster $\tau_{isotr} \approx 0.3 \,\rm fm/c$. The impact of the non-equilibrium implied by the saturation scale manifests for non-central collisions and can modify the estimate of the viscosity respect to the assumption of full thermalization in $p_T$-space. We find that the estimate of $\eta/s$ is modified from $\eta/s \approx 2/4\pi$ to $\eta/s \approx 1/4\pi$ at RHIC and from $\eta/s \approx 3/4\pi$ to $\eta/s \approx 2/4\pi$ at LHC. We complete our investigation by a study of the thermalization and isotropization times of the fireball for different initial conditions and values of $\eta/s$ showing how the latter affects both isotropization and thermalization. Lastly, we have seen that the range of values explored by the phase-space distribution function $f$ is such that at $p_T<0.5\, \rm GeV$ the inner part of the fireball stays with occupation number significantly larger than unity despite the fast longitudinal expansion, which might suggest the possibility of the formation of a transient Bose-Einstein Condensate.Comment: Published versio