https://slow-is-smooth.io/tags/column-generation/Recent content in Column Generation on Shai AsherHugoen-gbFri, 16 Oct 2026 08:00:00 +0300https://slow-is-smooth.io/blog/the-cleaning-robot-puzzle-planning-the-trips/Fri, 16 Oct 2026 08:00:00 +0300https://slow-is-smooth.io/blog/the-cleaning-robot-puzzle-planning-the-trips/<p><em>Part 3&rsquo;s fast plan empties the robot&rsquo;s bag on a furnished 12×12 room in 178 moves. The best plan takes 170, and this part proves that no plan takes fewer.</em></p> <p>This is <strong>Part 10</strong> of the cleaning robot puzzle. <a href="https://slow-is-smooth.io/blog/the-cleaning-robot-puzzle-four-new-rules/#emptying-the-bag">Part 3</a> gave the robot a small bag and a dock that empties it, and asked for the fewest moves that clean every dirty field. Its fast solver cuts one long tour into trips; its exact search tries every (field, cleaned fields, load) state and gives up beyond about nine dirty fields. <a href="https://slow-is-smooth.io/blog/the-cleaning-robot-puzzle-a-lower-bound-from-a-linear-program/">Parts 8</a> and <a href="https://slow-is-smooth.io/blog/the-cleaning-robot-puzzle-the-shortest-route-exactly/">9</a> wrote a route as numbers on pairs of fields. Here that model would need a whole copy of the room for every trip, so this part writes a plan the way the people who route real delivery vans do: as a <strong>choice among trips</strong>. There are more possible trips than atoms in the universe; the method below looks at a few hundred of them and still proves the answer. The new algorithms:</p>