Abstract: Motivated by the wavelength assignment problem in WDM optical networks, we study path coloring problems in graphs. Given a set of paths P on a graph G, the path coloring problem is to color the paths of P so that no two paths traversing the same edge of G are assigned the same color and the total number of colors used is minimized. The problem has been proved to be NP-hard even for trees and rings.
Using optimal solutions to fractional path coloring, a natural relaxation of path coloring, on which we apply a randomized rounding technique combined with existing coloring algorithms, we obtain new upper bounds on the minimum number of colors sufficient to color any set of paths on any graph. The upper bounds are either existential or constructive.
The existential upper bounds significantly improve existing ones provided that the cost of the optimal fractional path coloring is sufficiently large and the dilation of the set of paths is small. Our algorithmic results include improved approximation algorithms for path coloring in rings and in bidirected trees. Our results extend to variations of the original path coloring problem arizing in multifiber WDM optical networks.

Abstract: One of the most important features in image analysis and understanding is shape. Mathematical morphology is the image processing branch that deals with shape analysis. The definition of all morphological transformations is based on two primitive operations, i.e. dilation and erosion. Since many applications require the solution of morphological problems in real time, researching time efficient algorithms for these two operations is crucial.
In this paper, efficient algorithms for the binary as well as the grey level dilation and erosion are presented and evaluated for an advanced associative processor. It is shown through simulation results that the above architecture is near optimal in the binary case and is also as efficient as the array processor with a 2D-mesh interconnection in the grey level case. Finally, it is proven that the implementation of this image processing machine is economically feasible