Online square packing with prediction

Bin packing and its variants, such as online square packing, are classic optimization problems with wide-ranging applications in areas like virtual machine consolidation and supply chain management. This thesis investigates the online square packing problem, where the goal is to pack squares of various sizes into the minimum number of unit square bins. We assume the prediction model, which integrates potentially erroneous machine-learned predictions into online algorithms, offering insights about upcoming items in an input sequence. The primary focus of this thesis is to design algorithms that balance consistency (competitive ratio with accurate predictions) and robustness (competitive ratio under adversarial prediction errors), acknowledging the impact of prediction error on algorithm efficiency. This novel approach, diverging from traditional models with perfect foresight or static input distributions, incorporates the practical aspect of erroneous predictions into the study of online problems. The key contribution of this thesis is the development of \fullRap (\RAP), an online square packing algorithm with predictions, which achieves a consistency of $1.78$ and a robustness of $5.89$. \RAP utilizes predictions that are machine-learnable from a polynomial number of input sequence samples. Additionally, an extension of \RAP, \textsc{Adaptive-RAP}, is introduced. This sampling-based algorithm has an expected competitive ratio of at most $2.0885$, the best-known competitive ratio without predictions, and its competitive ratio approaches $1.78$, the consistency of \RAP, as more items are sampled. Furthermore, this work shows a lower bound on the robustness of any online classical bin packing algorithm for any consistency better than $1.3$; similarly, we establish a lower bound for the robustness of online square packing algorithms that have consistency better than $1.25$. These findings contribute to the understanding of the trade-offs between consistency and robustness in online packing problems under the prediction model.

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