FPT is Characterized by Useful Obstruction Sets

Many graph problems were first shown to be fixed-parameter tractable using the results of Robertson and Seymour on graph minors. We show that the combination of finite, computable, obstruction sets and efficient order tests is not just one way of obtaining strongly uniform FPT algorithms, but that all of FPT may be captured in this way. Our new characterization of FPT has a strong connection to the theory of kernelization, as we prove that problems with polynomial kernels can be characterized by obstruction sets whose elements have polynomial size. Consequently we investigate the interplay between the sizes of problem kernels and the sizes of the elements of such obstruction sets, obtaining several examples of how results in one area yield new insights in the other. We show how exponential-size minor-minimal obstructions for pathwidth k form the crucial ingredient in a novel OR-cross-composition for k-Pathwidth, complementing the trivial AND-composition that is known for this problem. In the other direction, we show that OR-cross-compositions into a parameterized problem can be used to rule out the existence of efficiently generated quasi-orders on its instances that characterize the NO-instances by polynomial-size obstructions.Comment: Extended abstract with appendix, as accepted to WG 201

Ten-dimensional wave packet simulations of methane scattering

We present results of wavepacket simulations of scattering of an oriented methane molecule from a flat surface including all nine internal vibrations. At a translational energy up to 96 kJ/mol we find that the scattering is almost completely elastic. Vibrational excitations when the molecule hits the surface and the corresponding deformation depend on generic features of the potential energy surface. In particular, our simulation indicate that for methane to dissociate the interaction of the molecule with the surface should lead to an elongated equilibrium C--H bond length close to the surface.Comment: RevTeX 15 pages, 3 eps figures: This article may be found at http://link.aip.org/link/?jcp/109/1966

Energy dissipation and scattering angle distribution analysis of the classical trajectory calculations of methane scattering from a Ni(111) surface

We present classical trajectory calculations of the rotational vibrational scattering of a non-rigid methane molecule from a Ni(111) surface. Energy dissipation and scattering angles have been studied as a function of the translational kinetic energy, the incidence angle, the (rotational) nozzle temperature, and the surface temperature. Scattering angles are somewhat towards the surface for the incidence angles of 30, 45, and 60 degree at a translational energy of 96 kJ/mol. Energy loss is primarily from the normal component of the translational energy. It is transfered for somewhat more than half to the surface and the rest is transfered mostly to rotational motion. The spread in the change of translational energy has a basis in the spread of the transfer to rotational energy, and can be enhanced by raising of the surface temperature through the transfer process to the surface motion.Comment: 8 pages REVTeX, 5 figures (eps

Ab-initio coupled-cluster effective interactions for the shell model: Application to neutron-rich oxygen and carbon isotopes

We derive and compute effective valence-space shell-model interactions from ab-initio coupled-cluster theory and apply them to open-shell and neutron-rich oxygen and carbon isotopes. Our shell-model interactions are based on nucleon-nucleon and three-nucleon forces from chiral effective-field theory. We compute the energies of ground and low-lying states, and find good agreement with experiment. In particular our calculations are consistent with the N=14, 16 shell closures in oxygen-22 and oxygen-24, while for carbon-20 the corresponding N=14 closure is weaker. We find good agreement between our coupled-cluster effective-interaction results with those obtained from standard single-reference coupled-cluster calculations for up to eight valence neutrons

Arithmetic Circuit Lower Bounds via MaxRank

We introduce the polynomial coefficient matrix and identify maximum rank of this matrix under variable substitution as a complexity measure for multivariate polynomials. We use our techniques to prove super-polynomial lower bounds against several classes of non-multilinear arithmetic circuits. In particular, we obtain the following results : As our main result, we prove that any homogeneous depth-3 circuit for computing the product of $d$ matrices of dimension $n \times n$ requires $\Omega(n^{d-1}/2^d)$ size. This improves the lower bounds by Nisan and Wigderson(1995) when $d=\omega(1)$. There is an explicit polynomial on $n$ variables and degree at most $\frac{n}{2}$ for which any depth-3 circuit $C$ of product dimension at most $\frac{n}{10}$ (dimension of the space of affine forms feeding into each product gate) requires size $2^{\Omega(n)}$. This generalizes the lower bounds against diagonal circuits proved by Saxena(2007). Diagonal circuits are of product dimension 1. We prove a $n^{\Omega(\log n)}$ lower bound on the size of product-sparse formulas. By definition, any multilinear formula is a product-sparse formula. Thus, our result extends the known super-polynomial lower bounds on the size of multilinear formulas by Raz(2006). We prove a $2^{\Omega(n)}$ lower bound on the size of partitioned arithmetic branching programs. This result extends the known exponential lower bound on the size of ordered arithmetic branching programs given by Jansen(2008).Comment: 22 page