Approximation techniques for unsplittable flow and traveling salesmen problems

Abstract

In this thesis, we present a variety of approximation algorithms for the Unsplittable Flow on Paths problem and some Traveling Salesman problems. The main contribution to the Unsplittable Flow on Paths problem is a logarithmic approximation algorithm which is the first non-trivial approximation for general instances of the problem. The algorithm works by using dynamic programming to approximate solutions on instances that cannot be approximated well through linear programming techniques. A generalization of this algorithm provides a constant-factor approximation in sub-exponential time. We also demonstrate that certain sparse instances can be approximated within a constant factor. The Traveling Salesman problems we consider mostly deal with finding paths in asymmetric metrics, though we do consider others. First, we demonstrate that the integrality gap of a natural linear programming relaxation for the Asymmetric Traveling Salesman Path problem is O(log n) where n is the number of nodes in the metric. We then further generalize the problem and study the problem of finding up to k paths with minimum total distance in an asymmetric metric such that the union of these paths spans all nodes. In the case that all paths are required to share a common start and end node, we demonstrate a family of bicriteria approximation algorithms that find a little more than k paths whose total cost is within some bounded ratio of the optimum value of a linear programming relaxation. These results are extended to many other variants of finding multiple paths in metrics whose union spans all nodes. However, we show that the most general case when each path has its own start and end location specified in advance cannot be approximated within any bounded ratio unless P = NP. Finally, we formulate a linear programming relaxation for the Minimum Latency problem in asymmetric metrics and prove that the integrality gap of this relaxation is O(log n). This critically relies on the fact that the integrality gap of a natural linear programming relaxation for the Asymmetric Traveling Salesman Path problem is O(log n). This is the first sub-polynomial approximation algorithm for the problem.