The Carpenter's Ruler Folding Problem

Summary

A carpenter's ruler is a ruler divided into pieces of different lengths which are hinged where the pieces meet, which makes it possible to fold the ruler. The carpenter's ruler folding problem, originally posed by Hopcroft, Joseph and Whitesides, is to determine the smallest case (or interval on the line) into which the ruler fits when folded. The problem is known to be NP-complete. The best previous approximation ratio achieved, dating from 1985, is 2. We improve this result and provide a fully polynomial-time approximation scheme for this problem. In contrast, in the plane, there exists a simple linear-time algorithm which computes an exact (optimal) folding of the ruler in some convex case of minimum diameter. This brings up the interesting problem of finding the minimum area of a convex universal case (of unit diameter) for all rulers whose maximum link length is one.

1. Introduction

The carpenter's ruler folding problem is: Given a sequence of rigid rods (links) of various integral lengths connected end-to-end by hinges, to fold it so that its overall folded length is minimum. It was first posed in [Hopcroft et al. 1985], where the authors proved that the problem is NP-complete using a reduction from the NP-complete problem PARTITION (see [Garey and Johnson 1979; Cormen et al. 1990]). A simple linear-time factor 2 approximation algorithm, as well as a pseudo-polynomial O﹛L²n) time dynamic programming algorithm, where L is the maximum link length, where presented in [Hopcroft et al. 1985] (see also [Kozen 1992]).