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11. A. Yao Separating the polynomial time hierarchy by oracles Proc. 26th Annu. IEEE Symp. Found. Comp. Sci. 1-10 (1985). Approximation Algorithms for 3D Orthogonal Knapsack⋆ Florian Diedrich1 , Rolf Harren2 , Klaus Jansen1 , Ralf Th¨ ole1 , and Henning Thomas1 1 Institut f¨ ur Informatik, Christian-Albrechts-Universit¨ at zu Kiel, Olshausenstr. 40, 24098 Kiel, Germany 2 Fachbereich Informatik, Universit¨ at Dortmund, Otto-Hahn-Str. 14, 44227 Dortmund, Germany Abstract. We study non-overlapping axis-parallel packings of 3D boxes with profits into a dedicated bigger box where rotation is forbidden; we wish to maximize the total profit.

Xdq , we have Wi′ (x1 , . . , xdq ) iff Wi ( x1 , . . , xd , . . , xd(q−1)+1 , . . , xdq ). We will use the notation S (ξ) = W0′ , . . , Wt′1 +t2 . The interpretation ξ is first-order definable in κ if all of the following relations are first-order definable in κ: W0′ , . . , Wt′1 +t2 . Definition 16. Suppose that W is a set of interpretations of L and d is a positive integer so that for all ξ in W , universe(ξ) ⊆ (universe(κ))d . We say that the set W is first-order definable in κ, if the set of sequences of relations {S (ξ) | ξ ∈ W } is first-order definable in κ, where S (ξ) is defined in the previous definition.

277-278. 9. A. M´ at´e, Nondeterministic polynomial-time computation and models of Arithmetic, Journal of the ACM, Vol. 37, (1), January 1990, pp. 175-193. 10. G. Takeuti, Proof Theory, North-Holland, Studies in Logic and the Foundations of Mathematics, Vol. 81, Second edition, 1987. 11. A. Yao Separating the polynomial time hierarchy by oracles Proc. 26th Annu. IEEE Symp. Found. Comp. Sci. 1-10 (1985). Approximation Algorithms for 3D Orthogonal Knapsack⋆ Florian Diedrich1 , Rolf Harren2 , Klaus Jansen1 , Ralf Th¨ ole1 , and Henning Thomas1 1 Institut f¨ ur Informatik, Christian-Albrechts-Universit¨ at zu Kiel, Olshausenstr.

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