Approximation algorithms for min-max resource sharing and malleable tasks scheduling

This thesis deals with approximation algorithms for problems in mathematical programming, combinatorial optimization, and their applications. We first study the Convex Min-Max Resource-Sharing problem (the Packing problem as the linear case) with M nonnegative convex constraints on a convex set B, which is a class of convex programming. In general block solvers are required for solving the problems. Based on a Lagrangian decomposition method via a logarithmic potential reduction, we generalize the algorithm by Grigoriadis et al. to the case with only weak approximate block solvers (i.e. with only constant, logarithmic or even worse approximation ratios). In this way we present an approximation algorithm for the Convex Min-Max Resource-Sharing problem that needs at most O(M(ln M+ε−2 ln ε−1)) calls to the block solver for any given relative accuracy ε ∈ (0, 1). It is the first bound independent of the data and the approximation ratio of the block solver for this general Convex Min-Max Resource-Sharing problem. In the case of small ratios we propose an improved approximation algorithm with at most O(M(ln M + ε−2)) calls to the block solver. As an application of the Convex Min-Max Resource-Sharing problem, we study the Multicast Congestion problem in communication networks. We are given a graph G = (V,E) to represent a communication network where |V | = n and |E| = m, and a set of multicast requests S1, . . . , Sk ⊆ V . A feasible solution to the Multicast Congestion problem in communication networks is a set of k trees T1, . . . , Tk where Ti connects the vertices in Si. The goal of the Multicast Congestion problem is to find a solution of k trees minimizing the maximum edge congestion (the number of times an edge is used). We develop a randomized asymptotic approximation algorithm for the Multicast Congestion problem in communication networks based on our approximation algorithm for the Convex