Ruler Rolling

At CCCG '21 O'Rourke proposed a variant of Hopcroft, Josephs and Whitesides' (1985) NP-complete problem {\sc Ruler Folding}, which he called {\sc Ruler Wrapping} and for which all folds must be 180 degrees in the same direction. Gagie, Saeidi and Sapucaia (2023) noted that if the last straight section of the ruler must be longest, then {\sc Ruler Wrapping} is equivalent to partitioning a string of positive integers into substrings whose sums are increasing such that the last substring sums to at most a given amount. They gave linear-time algorithms for the versions of {\sc Ruler Wrapping} both with and without this assumption. In real life we cannot repeatedly fold a carpenter's ruler 180 degrees in the same direction. In this paper we propose the more realistic problem of {\sc Ruler Rolling}, in which we repeatedly fold the segments 90 degrees in the same direction and thus fold the ruler into a rectangle instead of into an interval. We should report all the Pareto-optimal rollings. We note that if the last straight section of the ruler must be longer than the third to last -- analogously to Gagie et al.'s assumption -- then {\sc Ruler Rolling} is equivalent to partitioning a string of positive integers into substrings such that the sums of the even substrings are increasing, as are the sums of the odd substrings. We give a simple dynamic-programming algorithm that reports all the Pareto-optimal rollings in quadratic time under this assumption. Our algorithm still works even without the assumption, but then we are left with a quadratic number of two-dimensional feasible solutions, so finding the Pareto-optimal ones and increases our running time by a logarithmic factor. If we have a nice objective function, however, we still use quadratic time.