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Abstract:We study two fundamental three-dimensional (3D) geometric packing problems: 3D (Geometric) Bin Packing (3D-BP), and 3D Minimum Volume Bounding Box (3D-MVBB), where given a set of 3D (rectangular) cuboids, the goal is to find an axis-aligned nonoverlapping packing of all cuboids. In 3D-BP, we need to pack the given cuboids into the minimum number of unit cube bins. In 3D-MVBB, the goal is to pack them into a cuboid box of minimum volume.
It is NP-hard to even decide whether a set of rectangles can be packed into a unit square bin -- giving an (absolute) approximation hardness of $2$ for 3D-BP. The previous best (absolute) approximation for both the problems follows from a result of Buchwald and Scheithauer (Int.~Trans.~Oper.~Res., 2016), yielding approximation ratios of $11$, and $5+\varepsilon$, respectively, for 3D-BP and 3D-MVBB. We provide improved approximation ratios of $6$, and $3+\varepsilon$, respectively, for the two problems, for any constant $\varepsilon > 0$.
For 3D-BP, in the asymptotic regime, Bansal, Correa, Kenyon, and Sviridenko (Math.~Oper.~Res., 2006) showed that there is no asymptotic polynomial-time approximation scheme (APTAS) even when all items have the same height. Caprara (Math.~Oper.~Res., 2008) gave an asymptotic approximation ratio of $T_{\infty}^2 + \varepsilon\approx 2.86$, where $T_{\infty}$ is the well-known Harmonic constant in Bin Packing. We provide an algorithm with an improved asymptotic approximation ratio of $3T_{\infty}/2 +\varepsilon \approx 2.54$. Further, we show that unlike 3D-BP, 3D-MVBB admits an APTAS.

Submission history

From: Debajyoti Kar [view email]
[v1] Tue, 11 Mar 2025 20:07:42 UTC (252 KB)
[v2] Sat, 19 Apr 2025 09:53:48 UTC (253 KB)
[v3] Sat, 22 Aug 2026 15:41:11 UTC (330 KB)