Fast and Parallel Decomposition of Constraint Satisfaction Problems

Constraint Satisfaction Problems (CSP) are notoriously hard. Consequently, powerful decomposition methods have been developed to overcome this complexity. However, this poses the challenge of actually computing such a decomposition for a given CSP instance, and previous algorithms have shown their limitations in doing so. In this paper, we present a number of key algorithmic improvements and parallelisation techniques to compute so-called Generalized Hypertree Decompositions (GHDs) faster. We thus advance the ability to compute optimal (i.e., minimal-width) GHDs for a significantly wider range of CSP instances on modern machines. This lays the foundation for more systems and applications in evaluating CSPs and related problems (such as Conjunctive Query answering) based on their structural properties.

Universal Function Approximation by Deep Neural Nets with Bounded Width and ReLU Activations

This article concerns the expressive power of depth in neural nets with ReLU activations and a bounded width. We are particularly interested in the following questions: What is the minimal width w min ( d ) so that ReLU nets of width w min ( d ) (and arbitrary depth) can approximate any continuous function on the unit cube [ 0 , 1 ] d arbitrarily well? For ReLU nets near this minimal width, what can one say about the depth necessary to approximate a given function? We obtain an essentially complete answer to these questions for convex functions. Our approach is based on the observation that, due to the convexity of the ReLU activation, ReLU nets are particularly well suited to represent convex functions. In particular, we prove that ReLU nets with width d + 1 can approximate any continuous convex function of d variables arbitrarily well. These results then give quantitative depth estimates for the rate of approximation of any continuous scalar function on the d-dimensional cube [ 0 , 1 ] d by ReLU nets with width d + 3 .

Innovation in additive manufacturing of parts from aluminium matrix composites

The laser build-up cladding is a well-known technique for additive manufacturing tasks. Modern equipment for the laser cladding enables material to be deposited with the lateral resolution of about 100 μm and to manufacture miniature parts. In this paper the laser micro cladding process was investigated to produce miniature thin-wall parts of Al-based composites. Thin walls formation process by subsequent single tracks overlapping with vertical increment was investigated. The influence of the cladding parameters on the minimal width and the quality of the fabricated thin walls was examined. The thin walls with the minimal width of 140 μm and surface roughness Ra 1,5 μm were generated. Laser micro cladding potential to manufacture lattice-shaped structures of Al-Si composites was shown. Fabricated thin-wall structures can have application in different fields e.g. aviation, automotive and tooling industries.

Vertex Subsets with Minimal Width and Dual Width in $Q$-Polynomial Distance-Regular Graphs

We study $Q$-polynomial distance-regular graphs from the point of view of what we call descendents, that is to say, those vertex subsets with the property that the width $w$ and dual width $w^*$ satisfy $w+w^*=d$, where $d$ is the diameter of the graph. We show among other results that a nontrivial descendent with $w\geq 2$ is convex precisely when the graph has classical parameters. The classification of descendents has been done for the $5$ classical families of graphs associated with short regular semilattices. We revisit and characterize these families in terms of posets consisting of descendents, and extend the classification to all of the $15$ known infinite families with classical parameters and with unbounded diameter.

A Note about Bezdek's Conjecture on Covering an Annulus by Strips

A closed plane region between two parallel lines is called a strip. András Bezdek posed the following conjecture: For each convex region $K$ there is an $\varepsilon>0$ such that if $\varepsilon K$ lies in the interior of $K$ and the annulus $K\backslash \varepsilon K$ is covered by finitely many strips, then the sum of the widths of the strips must be at least the minimal width of $K$. In this paper, we consider problems which are related to the conjecture.

Some inequalities for planar convex sets containing one lattice point

We obtain two inequalities relating the diameter and the (minimal) width with the area of a planar convex set containing exactly one point of the integer lattice in its interior. They are best possible. We then use these results to obtain some related inequalities.