Discovering and Certifying Lower Bounds for the Online Bin Stretching Problem

Abstract

There are several problems in the theory of online computation where tight lower bounds on the competitive ratio are unknown and expected to be difficult to describe in a short form. A good example is the Online Bin Stretching problem, in which the task is to pack the incoming items online into bins while minimizing the load of the largest bin. Additionally, the optimal load of the entire instance is known in advance. The contribution of this paper is twofold. First, we provide the first non-trivial lower bounds for Online Bin Stretching with 6, 7 and 8 bins, and increase the best known lower bound for 3 bins. We describe in detail the algorithmic improvements which were necessary for the discovery of the new lower bounds, which are several orders of magnitude more complex. The lower bounds are presented in the form of directed acyclic graphs. Second, we use the Coq proof assistant to formalize the Online Bin Stretching problem and certify these large lower bound graphs. The script we propose certified as well all the previously claimed lower bounds, which until now were never formally proven. To the best of our knowledge, this is the first use of a formal verification toolkit to certify a lower bound for an online problem