Complexity of the Stamp Folding Problem

Abstract

For a given mountain-valley pattern of equidistant creases on a long paper strip, there are many folded states consistent with the pattern. Among these folded states, we like to fold a paper so that the number of the paper layers between each pair of hinged paper segments, which is called the crease width at the crease point, is minimized. This problem is called the stamp folding problem and there are two variants of this problem; minimization of the maximum crease width, and minimization of the total crease width. This optimization problem is recently introduced and investigated from the viewpoint of the counting problem. However, its computational complexity is not known. In this paper, we first show that the minimization problem of the maximum crease width is strongly NP-complete. Hence we cannot solve the problem in polynomial time unless P=NP. We next propose an algorithm that solves the minimization problem. The algorithm itself is a straightforward one, but its analysis is not trivial. We show that this algorithm runs in $O{\left({n^2 {{n+k}\choose{k}}}\right)}$ time where n is the number of creases and k is the total crease width. That is, the algorithm runs in O(n ^k + 2) time for a constant k. Hence we can solve the problem efficiently for a small constant k.