Local edge colouring of Yao-like subgraphs of Unit Disk Graphs

The focus of the present paper is on providing a local deterministic algorithm for colouring the edges of Yao-like   subgraphs of Unit Disk Graphs. These are geometric graphs such that for some positive integers l,k$l,k$ the following property holds at each node v$v$: if we partition the unit circle centered at v$v$ into 2k$2k$ equally sized wedges then each wedge can contain at most l$l$ points different from v$v$. We assume that the nodes are location aware, i.e. they know their Cartesian coordinates in the plane. The algorithm presented is local in the sense that each node can receive information emanating only from nodes which are at most a constant (depending on k$k$ and l$l$, but not on the size of the graph) number of hops (measured in the original underlying Unit Disk Graph) away from it, and hence the algorithm terminates in a constant number of steps. The number of colours used is 2kl+1$2kl+1$ and this is optimal for local algorithms (since the maximal degree is 2kl$2kl$ and a colouring with 2kl$2kl$ colours can only be constructed by a global algorithm), thus showing that in this class of graphs the price for locality is only one additional colour.     

Divergence bounded computable real numbers

A real x$x$ is called h$h$-bounded computable  , for some function h:N→N$h:\mathbb{N}\to \mathbb{N}$, if there is a computable sequence (_xs)$(x_s)$ of rational numbers which converges to x$x$ such that, for any n∈N$n\in \mathbb{N}$, at most h(n)$h(n)$ non-overlapping pairs of its members are separated by a distance larger than 2^-n$2^{-n}$. In this paper we discuss properties of h$h$-bounded computable reals for various functions h$h$. We will show a simple sufficient condition for a class of functions h$h$ such that the corresponding h$h$-bounded computable reals form an algebraic field. A hierarchy theorem for h$h$-bounded computable reals is also shown. Besides we compare semi-computability and weak computability with the h$h$-bounded computability for special functions h$h$.     

Rational two-parameter families of spheres and rational offset surfaces

The present paper investigates two-parameter families of spheres in R³${\mathbb{R}}^3$ and their corresponding two-dimensional surfaces Φ$\Phi$ in R⁴${\mathbb{R}}^4$. Considering a rational surface Φ$\Phi$ in R⁴${\mathbb{R}}^4$, the envelope surface Ψ$\Psi$ of the corresponding family of spheres in R³${\mathbb{R}}^3$ is typically non-rational. Using a classical sphere-geometric approach, we prove that the envelope surface Ψ$\Psi$ and its offset surfaces admit rational parameterizations if and only if Φ$\Phi$ is a rational sub-variety of a rational isotropic hyper-surface in R⁴${\mathbb{R}}^4$. The close relation between the envelope surfaces Ψ$\Psi$ and rational offset surfaces in R³${\mathbb{R}}^3$ is elaborated in detail. This connection leads to explicit rational parameterizations for all rational surfaces Φ$\Phi$ in R⁴${\mathbb{R}}^4$ whose corresponding two-parameter families of spheres possess envelope surfaces admitting rational parameterizations. Finally we discuss several classes of surfaces sharing this property.     

Quadratic programming with transaction costs

We consider the problem of maximizing the mean-variance utility function of n$n$ assets. Associated with a change in an asset's holdings from its current or target value is a transaction cost. These must be accounted for in practical problems. A straightforward way of doing so results in a 3n$3n$-dimensional optimization problem with 3n$3n$ additional constraints. This higher dimensional problem is computationally expensive to solve. We present an algorithm for solving the 3n$3n$-dimensional problem by modifying an active set quadratic programming (QP) algorithm to solve the 3n$3n$-dimensional problem as an n$n$-dimensional problem accounting for the transaction costs implicitly rather than explicitly. The method is based on deriving the optimality conditions for the higher dimensional problem solely in terms of lower dimensional quantities and requires substantially less computational effort than any active set QP algorithm applied directly on a 3n$3n$-dimensional problem.     

Transposition invariant words

We define an operation called transposition on words of fixed length. This operation arises naturally when the letters of a word are considered as entries of a matrix. Words that are invariant with respect to transposition are of special interest. It turns out that transposition invariant words have a simple interpretation by means of elementary group theory. This leads us to investigate some properties of the ring of integers modulo n$n$ and primitive roots. In particular, we show that there are infinitely many prime numbers p$p$ with a primitive root dividing p+1$p+1$ and infinitely many prime numbers p$p$ without a primitive root dividing p+1$p+1$. We also consider the orbit of a word under transposition.     

Paths and trails in edge-colored graphs

This paper deals with the existence and search for properly edge-colored paths/trails between two, not necessarily distinct, vertices s$s$ and t$t$ in an edge-colored graph from an algorithmic perspective. First we show that several versions of the s−t$s-t$ path/trail problem have polynomial solutions including the shortest path/trail case. We give polynomial algorithms for finding a longest properly edge-colored path/trail between s$s$ and t$t$ for a particular class of graphs and characterize edge-colored graphs without properly edge-colored closed trails. Next, we prove that deciding whether there exist k$k$ pairwise vertex/edge disjoint properly edge-colored s−t$s-t$ paths/trails in a c$c$-edge-colored graph G^c$G^c$ is NP-complete even for k=2$k=2$ and c=Ω(n²)$c=\Omega \left(n^2\right)$, where n$n$ denotes the number of vertices in G^c$G^c$. Moreover, we prove that these problems remain NP-complete for c$c$-edge-colored graphs containing no properly edge-colored cycles and c=Ω(n)$c=\Omega \left(n\right)$. We obtain some approximation results for those maximization problems together with polynomial results for some particular classes of edge-colored graphs.     

Spanners for bounded tree-length graphs

This paper concerns construction of additive stretched spanners with few edges for n$n$-vertex graphs having a tree-decomposition into bags of diameter at most δ$\delta$, i.e., the tree-length δ$\delta$ graphs. For such graphs we construct additive 2δ$2\delta$-spanners with O(δn+nlogn)$O(\delta n+nlogn)$ edges, and additive 4δ$4\delta$-spanners with O(δn)$O\left(\delta n\right)$ edges. This provides new upper bounds for chordal graphs for which δ=1$\delta =1$. We also show a lower bound, and prove that there are graphs of tree-length δ$\delta$ for which every multiplicative δ$\delta$-spanner (and thus every additive (δ−1)$(\delta -1)$-spanner) requires Ω(n^1+1/Θ(δ))$\Omega \left(n^{1+1/\Theta \left(\delta \right)}\right)$ edges.