In this paper we establish a general algorithmic framework between bin packing and strip packing, with which we achieve the same asymptotic bounds by applying bin packing algorithms to strip packing. More precisely we obtain the following results: (1) Any offline bin packing algorithm can be applied to strip packing maintaining the same asymptotic worst-case ratio. Thus using FFD (MFFD) as a su...

The vector bin packing problem (VBP) is a generalization of bin packing with multiple constraints. In this problem we are required to pack items, represented by p-dimensional vectors, into as few bins as possible. The multiple-choice vector bin packing (MVBP) is a variant of the VBP in which bins have several types and items have several incarnations. We present an exact method, based on an arc...

Let {X(t)}t≥0 denote a Lévy process in R with exponent Ψ. Taylor (1986) proved that the packing dimension of the range X([0 , 1]) is given by the index (0.1) γ′ = sup { α ≥ 0 : lim inf r→0+ ∫ 1 0 P {|X(t)| ≤ r} rα dt = 0 } . We provide an alternative formulation of γ′ in terms of the Lévy exponent Ψ. Our formulation, as well as methods, are Fourier-analytic, and rely on the properties of the Ca...

In the context of Kolmogorov’s algorithmic approach to the foundations of probability, Martin-Löf defined the concept of an individual random sequence using the concept of a constructive measure 1 set. Alternate characterizations use constructive martingales and measures of impossibility. We prove a direct conversion of a constructive martingale into a measure of impossibility and vice versa, s...

We shall use the \iterated Venetian blind" construction to show that the packing dimensions of plane sections of subsets of R n can depend essentially on the directions of the planes. We shall also establish the instability of the packing dimension of sections under smooth dieomorphisms.