The Resource Allocation Problems seek to find an optimal repartition of resources into a fixed number of areas. In this thesis, we consider a resource allocation problem with a linear objective and two distinct sets of constraints: a set of nested constraints, where the partial sums of the decision variables are limited from above and a linear constraint that defines a hyperplane. We propose a weakly and a strongly polynomial algorithm. The weakly polynomial algorithm requires certain assumptions of the data and runs in O(n log n log |Λ|/|I|) time, where n is the number of decision variables, Λ is an interval in the dual space, and |I| relates to the precision of the data. The strongly polynomial algorithm is based on Megiddo s parametric search technique, and obtains a complexity of O(n log n). These are large improvements upon the O(n 3/ log n) complexity of the generic Interior Point Method. In addition, an experimental analysis was carried out and the algorithms showed to be more efficient and produced optimal solutions for problem instances with up to 1,000,000 variables.