An Optimal Algorithm for 1-D Cutting Stock Problem

We present an $n{\mathrm{\Delta }}^{O(k^2)}$ time algorithm to obtain an optimal solution for $1$-dimensional cutting stock problem: the bin packing problem of packing $n$ items onto unit capacity bins under the restriction that the number of item sizes $k$ is fixed, where $\mathrm{\Delta }$ is the reciprocal of the size of the smallest item. We employ elementary ideas in both the design and analysis our algorithm.