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In 1985 Hopcroft, Joseph and Whitesides showed it is NP-complete to decide whether a carpenter’s ruler with segments of given positive lengths can be folded into an interval of at most a given length, such that the folded hinges alternate between 180 degrees clockwise and 180 degrees counter-clockwise. At the open-problem session of 33rd Canadian Conference on Computational Geometry (CCCG ’21), O’Rourke proposed a natural variation of this problem called ruler wrapping, in which all folded hinges must be folded the same way. In this paper we show O’Rourke’s variation has a linear-time solution.

We propose a natural variation of the Hopcroft, Joseph and Whitesides ' (1985) classic problem of folding a carpenter's ruler into the shortest possible interval. In the original problem folds must alternate between 180 degrees clockwise and 180 degrees counterclockwise but in our version, which we call ruler wrapping, all the folds must be in the same direction. Whereas the original problem is NP-complete, we first show our version has a simple and fairly obvious quadratic-time algorithm, and then show it has a somewhat less obvious linear-time algorithm which is nevertheless still simple enough to be implemented in under a page of C code.

The partition of a problem into smaller sub-problems satisfying certain properties is often a key ingredient in the design of divide-and-conquer algorithms. For questions related to location, the partition problem can be modeled, in geometric terms, as finding a subdivision of a planar map -which represents, say, a geographical area -into regions subject to certain conditions while optimizing some objective function. In this paper, we investigate one of these geometric problems known as the Minimum Convex Partition Problem (mcpp). A convex partition of a point set P in the plane is a subdivision of the convex hull of P whose edges are segments with both endpoints in P and such that all internal faces are empty convex polygons. The mcpp is an NP-hard problem where one seeks to find a convex partition with the least number of faces.We present a novel polygon-based integer programming formulation for the mcpp, which leads to better dual bounds than the previously known edge-based model. Moreover, we introduce a primal heuristic, a branching rule and a pricing algorithm. The combination of these techniques leads to the ability to solve instances with twice as many points as previously possible while constrained to identical computational resources. A comprehensive experimental study is presented to show the impact of our design choices.