High Multiplicity Strip Packing with Three Rectangle Types

Abstract

Abstract The two-dimensional strip packing problem consists of packing in a rectangular strip of width \(1\) and minimum height a set of \(n\) rectangles, where each rectangle has width \(0 < w \leq 1\) and height \(0 < h \leq h_{max}\). We consider the high-multiplicity version of the problem in which there are only \(K\) different types of rectangles. For the case when \(K = 3\), we give an algorithm providing a solution requiring at most height \(\frac{3}{2}h_{max} + \epsilon\) plus the height of an optimal solution, where \(\epsilon\) is any positive constant. For the case when \(K = 4\), we give an algorithm providing a solution requiring at most \(\frac{7}{3}h_{max} + \epsilon\) plus the height of an optimal solution. For the case when \(K > 3\), we give an algorithm providing a solution requiring at most \(\lfloor\frac{3}{4}K\rfloor + 1 + \epsilon\) plus the height of an optimal solution.