title: LP-based approximation algorithms for partial-ordered scheduling and matroid augmentation
Creators: name: Stickler, Karl Christoph Heinz Walter
Publisher: name: London School of Economics and Political Science (LSE)
Year: 2023

description: In this thesis, we study two NP-hard problems from Combinatorial Optimization, from the perspective of approximation algorithms. The first problem we study is called Partial-Order Scheduling on Parallel Machines, which we abbreviate to PO Scheduling. Here, we are given a partially ordered set of jobs which we want to schedule to a set of machines. Each job has some weight and some processing time associated to it. On each machine, the order of the jobs scheduled to it must agree with the given partial order, i.e., a job can only be started once all its predecessors scheduled to the same machine have been completed. However, two jobs scheduled to different machines are not constrained in any way. Thus, PO Scheduling deviates from the well-studied problem of precedence-constrained scheduling in this regard. The goal of PO Scheduling is to find a feasible schedule which minimizes the sum of weighted completion times of the jobs. PO Scheduling generalizes an already NP-hard version of scheduling introduced by Bosman, Frascaria, Olver, Sitters and Stougie [3], where they ask the same question as in PO Scheduling for the case where the jobs are totally ordered. The authors above present a constant-factor approximation algorithm for their problem. We conjecture that there is a constant-factor approximation algorithm for PO Scheduling as well. While we do not solve the problem, we give approximation algorithms for the special case that the partial order consists of disjoint totally ordered chains of linearly bounded length. Additionally, we give a structural result for optimal schedules in the case that the partial order consists of disjoint, backwardly ordered (with regard to the Smith-ratio) chains. We point towards some potential research directions. For the Weighted Tree Augmentation Problem, we are given a graph with a distinguished spanning tree. Each non tree-edge has a cost associated to it. The goal is to find a cost-minimal set of edges such that when we add them to the tree-edges, the resulting graph is 2-edge-connected. Weighted tree augmentation is NP-hard. There has been recent progress in decreasing the best-known approximation factor for the problem by Traub and Zenklusen to (1.5 + ε) [51, 52]. We study a generalization of weighted tree augmentation, called the Weighted Matroid Augmentation Problem, which we abbreviate to WMAP. In WMAP, we consider a matroid with a distinguished basis and a cost function on the non-basis elements. The goal is to find a cost-minimal set such that the union of the fundamental circuits of the elements in the set with regard to the distinguished basis cover that basis. We conjecture that there is a 2-approximation algorithm for the problem in the case that the matroid is regular. While we do not solve the problem, we give an approximation algorithm for the special case of the cographic matroid and show that there is no constant-factor approximation algorithm for WMAP for representable matroids unless P = NP.