Approximation Algorithms for Stochastic Inventory Control Models

Approximation Algorithms for Stochastic Inventory Control Model

Development of mental transformation abilities

Mental representation and transformation of spatial information is often examined with mental rotation tasks, which require deciding whether a rotated image is the same or the mirror version of an upright image. Recent research with infants shows early discrimination of objects from mirror image versions. However, even at age 4, many children perform near chance level on more standard measures. Similar age discrepancies can be observed in other domains, including perspective taking, theory of mind, and intuitive physics. These paradoxical results raise the questions of how performance relates to competence, and how to conceptualize developmental change. There may be a common underlying mechanism: the development of the ability to imagine things and mentally transform them in a prospective fashion

Using mental transformation strategies for spatial scaling: Evidence from a discrimination task

Spatial scaling, or an understanding of how distances in different-sized spaces relate to each other, is fundamental for many spatial tasks and relevant for success in numerous professions. Previous research has suggested that adults use mental transformation strategies to mentally scale spatial input, as indicated by linear increases in response times and accuracies with larger scaling magnitudes. However, prior research has not accounted for possible difficulties in encoding spatial information within smaller spaces. Thus, the present study used a discrimination task in which we systematically pitted absolute size of the spaces against scaling magnitude. Adults (N = 48) were presented with 2 pictures, side-by-side on a computer display, each of which contained a target. Adults were asked to decide whether the targets were in the same position or not, by pressing the respective computer key. In the constant-large condition, the constant space was kept large, whereas the size of the other space was variable and smaller. In the constant-small condition, the constant space was small, whereas the size of the other space was variable and larger. Irrespective of condition, adults’ discrimination performance (d- primes) and response times were linear functions of scaling magnitude, supporting the notion that analog imagery strategies are used in spatial scaling

A Note on Scheduling Problems with Irregular Starting Time Costs

In [9], Maniezzo and Mingozzi study a project scheduling problem with irregular starting time costs. Starting from the assumption that its computational complexity status is open, they develop a branch-and-bound procedure, and identify special cases that are solvable in polynomial time. In this note, we review three previously established, related results which show that the general problem is solvable in polynomial time

Density of critical points for a Gaussian random function

Critical points of a scalar quantitiy are either extremal points or saddle points. The character of the critical points is determined by the sign distribution of the eigenvalues of the Hessian matrix. For a two-dimensional homogeneous and isotropic random function topological arguments are sufficient to show that all possible sign combinations are equidistributed or with other words, the density of the saddle points and extrema agree. This argument breaks down in three dimensions. All ratios of the densities of saddle points and extrema larger than one are possible. For a homogeneous Gaussian random field one finds no longer an equidistribution of signs, saddle points are slightly more frequent.Comment: 11 pages 1 figure, changes in list of references, corrected typo

On the stable degree of graphs

We define the stable degree s(G) of a graph G by s(G)∈=∈ min max d (v), where the minimum is taken over all maximal independent sets U of G. For this new parameter we prove the following. Deciding whether a graph has stable degree at most k is NP-complete for every fixed k∈≥∈3; and the stable degree is hard to approximate. For asteroidal triple-free graphs and graphs of bounded asteroidal number the stable degree can be computed in polynomial time. For graphs in these classes the treewidth is bounded from below and above in terms of the stable degree

On Sparsification for Computing Treewidth

We investigate whether an n-vertex instance (G,k) of Treewidth, asking whether the graph G has treewidth at most k, can efficiently be made sparse without changing its answer. By giving a special form of OR-cross-composition, we prove that this is unlikely: if there is an e > 0 and a polynomial-time algorithm that reduces n-vertex Treewidth instances to equivalent instances, of an arbitrary problem, with O(n^{2-e}) bits, then NP is in coNP/poly and the polynomial hierarchy collapses to its third level. Our sparsification lower bound has implications for structural parameterizations of Treewidth: parameterizations by measures that do not exceed the vertex count, cannot have kernels with O(k^{2-e}) bits for any e > 0, unless NP is in coNP/poly. Motivated by the question of determining the optimal kernel size for Treewidth parameterized by vertex cover, we improve the O(k^3)-vertex kernel from Bodlaender et al. (STACS 2011) to a kernel with O(k^2) vertices. Our improved kernel is based on a novel form of treewidth-invariant set. We use the q-expansion lemma of Fomin et al. (STACS 2011) to find such sets efficiently in graphs whose vertex count is superquadratic in their vertex cover number.Comment: 21 pages. Full version of the extended abstract presented at IPEC 201

Kernel Bounds for Structural Parameterizations of Pathwidth

Assuming the AND-distillation conjecture, the Pathwidth problem of determining whether a given graph G has pathwidth at most k admits no polynomial kernelization with respect to k. The present work studies the existence of polynomial kernels for Pathwidth with respect to other, structural, parameters. Our main result is that, unless NP is in coNP/poly, Pathwidth admits no polynomial kernelization even when parameterized by the vertex deletion distance to a clique, by giving a cross-composition from Cutwidth. The cross-composition works also for Treewidth, improving over previous lower bounds by the present authors. For Pathwidth, our result rules out polynomial kernels with respect to the distance to various classes of polynomial-time solvable inputs, like interval or cluster graphs. This leads to the question whether there are nontrivial structural parameters for which Pathwidth does admit a polynomial kernelization. To answer this, we give a collection of graph reduction rules that are safe for Pathwidth. We analyze the success of these results and obtain polynomial kernelizations with respect to the following parameters: the size of a vertex cover of the graph, the vertex deletion distance to a graph where each connected component is a star, and the vertex deletion distance to a graph where each connected component has at most c vertices.Comment: This paper contains the proofs omitted from the extended abstract published in the proceedings of Algorithm Theory - SWAT 2012 - 13th Scandinavian Symposium and Workshops, Helsinki, Finland, July 4-6, 201

The relation between spatial thinking and proportional reasoning in preschoolers

Previous research has indicated a close link between spatial and mathematical thinking. However, what shared processes account for this link? In this study, we focused on the spatial skill of map reading and the mathematical skill of proportional reasoning and investigated whether scaling, or the ability to relate information in different-sized representations, is a shared process. Scaling was experimentally manipulated in both tasks. In the map task, 4- and 5-year-olds (N = 50) were asked to point to the same position shown on a map in a larger referent space on a touch screen. The sizes of the maps were varied systematically, such that some trials required scaling and some did not (i.e., the map had the same size as the referent space). In the proportional reasoning task, children were presented with different relative amounts of juice and water and were asked to estimate each mixture on a rating scale. Again, some trials required scaling, but others could be solved by directly mapping the proportional components onto the rating scale. Children’s absolute errors in locating targets in the map task were closely related to their performance in the proportional reasoning task even after controlling for age and verbal intelligence. Crucially, this was only true for trials that required scaling, whereas performance on nonscaled trials was not related. These results shed light on the mechanisms involved in the close connection between spatial and mathematical thinking early in life

Molecular dynamics approach: from chaotic to statistical properties of compound nuclei

Statistical aspects of the dynamics of chaotic scattering in the classical model of $\alpha$-cluster nuclei are studied. It is found that the dynamics governed by hyperbolic instabilities which results in an exponential decay of the survival probability evolves to a limiting energy distribution whose density develops the Boltzmann form. The angular distribution of the corresponding decay products shows symmetry with respect to $\pi/2$ angle. Time estimated for the compound nucleus formation ranges within the order of $10^{-21}$s.Comment: 11 pages, LaTeX, non