On the significance of information in online decision-making : explored in online load balancing and in buffer minimization with conflicts

When making decisions, the available information plays a crucial role. In online problems, decisions must be made sequentially as information is revealed over time. This means that early choices can influence or limit future options. In this thesis we examine the impact of information in two online problems: bin stretching and the buffer minimization problem with conflicts. To assess the quality of a decision, we apply the worst-case measure known as the competitive ratio, a tool of competitive analysis. It compares the outcome of an online decision—made with partial information—to an optimal decision with complete knowledge in hindsight. For both problems we study the effect of providing the value of the optimal offline solution’s outcome in advance on the competitive ratio. The value serves as a reference scale.
Bin stretching is an online load balancing problem in which items arrive one by one. For each item we must irrevocably decide in which bin we pack the item, without knowledge of future items. Given the information that all items can be packed in unit-sized bins, the goal is to minimize the maximum load over all bins. We propose a sophisticated two-phase algorithm that surpasses the natural barrier of 3/2, as long as sufficiently many bins are available. The rigorous analysis of its Performance consists of an intricate mixture of size and weight arguments. Further contributions include valuable insights into the problem’s difficulties, extending beyond the known results for smaller numbers of bins, which are based on computer search. We present various structural design constraints that any effective algorithm must adhere to and outline directions for future improvements.
The buffer minimization problem with conflicts is an online scheduling Problem in which machines share a common resource. Jobs arrive sequentially but separately on the machines. Once a job is revealed its load is stored in the machine’s Input buffer. Processing a job reduces the buffer load, but conflicts between machines restrict simultaneous execution. These conflicts are modeled by a conflict graph, where an edge between a pair of machines indicates a conflict. The objective is to provide a valid schedule that respects the conflict constraints with the goal of minimizing the maximum load that is ever stored in a single buffer. A priori, the algorithm is given the information that unit-sized buffers are sufficient to process the complete input. We study the problem in the recently introduced flow model, where loads arrive as continuous flow rather than in discrete blocks. In this setting, we present tight bounds for all conflict-graphs with four vertices, except the path which has previously already been resolved, and for the family of complete graphs. For complete bipartite graphs, we recover almost tight bounds.