5 research outputs found
Online Bin Stretching with Three Bins
Online Bin Stretching is a semi-online variant of bin packing in which the
algorithm has to use the same number of bins as an optimal packing, but is
allowed to slightly overpack the bins. The goal is to minimize the amount of
overpacking, i.e., the maximum size packed into any bin.
We give an algorithm for Online Bin Stretching with a stretching factor of
for three bins. Additionally, we present a lower bound of for Online Bin Stretching on three bins and a lower bound of
for four and five bins that were discovered using a computer search.Comment: Preprint of a journal version. See version 2 for the conference
paper. Conference paper split into two journal submissions; see
arXiv:1601.0811
Discovering and Certifying Lower Bounds for the Online Bin Stretching Problem
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
Improved Lower Bounds for the Online Bin~Stretching Problem
We use game theory techniques to automatically compute improved lower bounds on the competitive ratio for the bin stretching problem. Using these techniques, we improve the best lower bound for this problem to 19/14. We explain the technique and show that it can be generalized to compute lower bounds for any online or semi-online packing or scheduling problem. We also present a lower bound, with value 7/6, on the expected competitive ratio of randomized algorithms for the bin stretching problem
Improved lower bounds for the online bin stretching problem
International audienceWe use game theory techniques to automatically compute improved lowerbounds on the competitive ratio for the bin stretching problem. Using these techniques,we improve the best lower bound for this problem to 19/14. We explain the techniqueand show that it can be generalized to compute lower bounds for any online or semi-online packing or scheduling problem