Semi-online models for cardinality constrained bin packing

Abstract We study two semi-online models for bin packing and exhibit them on cardinality constrained bin packing with small values of k . In this variant of the bin packing problem, each bin can have at most k items whose total size does not exceed 1. For the semi-online model where the algorithm may use a reordering buffer, we show that even if a single item can be stored in the buffer at any point in time, the best possible asymptotic competitive ratio for the case $$k=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> is smaller than that of the purely online problem. For the model with two parallel solutions, which is equivalent to the model with advice with a single bit of advice, we show an improved upper bound on the asymptotic competitive ratio for $$k=3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> .