About equivalent interval colorings of weighted graphs

Given a graph G=(V,E) with strictly positive integer weights ω_i on the vertices iV, a k-interval coloring of G is a function I that assigns an interval I(i){1,…,k} of ω_i consecutive integers (called colors) to each vertex iV. If two adjacent vertices x and y have common colors, i.e. I(i)∩I(j)≠0/ for an edge [i,j] in G, then the edge [i,j] is said conflicting. A k-interval coloring without conflicting edges is said legal. The interval coloring problem (ICP) is to determine the smallest integer k, called interval chromatic number of G and denoted χ_int(G), such that there exists a legal k-interval coloring of G. For a fixed integer k, the k-interval graph coloring problem (k-ICP) is to determine a k-interval coloring of G with a minimum number of conflicting edges. The ICP and k-ICP generalize classical vertex coloring problems where a single color has to be assigned to each vertex (i.e., ω_i=1 for all vertices iV).Two k-interval colorings I₁ and I₂ are said equivalent if there is a permutation π of the integers 1,…,k such that ℓI₁(i) if and only if π(ℓ)I₂(i) for all vertices iV. As for classical vertex coloring, the efficiency of algorithms that solve the ICP or the k-ICP can be increased by avoiding considering equivalent k-interval colorings, assuming that they can be identified very quickly. To this purpose, we define and prove a necessary and sufficient condition for the equivalence of two k-interval colorings. We then show how a simple tabu search algorithm for the k-ICP can possibly be improved by forbidding the visit of equivalent solutions.     

Isotropic affine spheres

In this paper, we study affine spheres which are isotropic and we obtain a complete classification. In particular, we show that all such affine spheres are hyperbolic affine spheres, isometric with SL(3 , R ) / SO(3), SL(3 , C ) / SU(3), SU*(6) / Sp(3) or E6 (-26) /F4 .     

Data mining usage in Italian SMEs: an integrated SEM-ANN approach

Central European Journal of Operations Research - Data mining is the process of knowledge extraction from the data with the algorithms that identify hidden relationships and patterns, which are...     

The numerical methods in the field of operators

The approximate solution of a class of nonlinear differential equations, in the field of Mikusiński operators is constructed by using the Euler method. The obtained results are applied to a class of partial integro–differential equations. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)