https://slow-is-smooth.io/tags/linear-programming/Recent content in Linear Programming on Shai AsherHugoen-gbFri, 02 Oct 2026 08:00:00 +0300https://slow-is-smooth.io/blog/the-cleaning-robot-puzzle-a-lower-bound-from-a-linear-program/Fri, 02 Oct 2026 08:00:00 +0300https://slow-is-smooth.io/blog/the-cleaning-robot-puzzle-a-lower-bound-from-a-linear-program/<p><em>Part 3&rsquo;s fast route cleans a furnished 40×40 room in 1,327 commands, and all Part 3 could prove is that no route needs fewer than 1,037. Between the two lie 290 commands nobody could account for.</em></p> <p>This is <strong>Part 8</strong> of the cleaning robot puzzle, and the first of the parts that take the robot&rsquo;s optimisation problems to the tools operations research really uses. <a href="https://slow-is-smooth.io/blog/the-cleaning-robot-puzzle-four-new-rules/#fewest-commands">Part 3</a> asked for the fewest commands that clean the whole floor, built a fast solver, an exact one that gives up on big rooms, and a lower bound from three simple facts. It ended on an open question: &ldquo;On big furnished rooms the gap to the lower bound stays wide. The bound is weak there, and nobody knows how far from the best the fast routes really are.&rdquo;</p>