LP Bounds in an Interval-Graph Algorithm for Orthogonal-Packing Feasibility
研究了高维正交装箱可行性问题，通过一维条形线性规划松弛和区间图算法，利用物品投影重叠信息收紧松弛，数值实验显示算法加速和稳定。
We consider the feasibility problem OPP (orthogonal packing problem) in higher-dimensional orthogonal packing: given a set of d-dimensional (d ≥ 2) rectangular items, decide whether all of them can be orthogonally packed in the given rectangular container without rotation. The one-dimensional (1D) “bar” LP relaxation of OPP reduces the latter to a 1D cutting-stock problem where the packing of each stock bar represents a possible 1D stitch through an OPP layout. The dual multipliers of the LP provide us with another kind of powerful bounding information (conservative scales). We investigate how the set of possible 1D packings can be tightened using the overlapping information of item projections on the axes, with the goal to tighten the relaxation. We integrate the bar relaxation into an interval-graph algorithm for OPP, which operates on such overlapping relations. Numerical results on 2D and 3D instances demonstrate the efficiency of tightening leading to a speedup and stabilization of the algorithm.