https://slow-is-smooth.io/tags/integer-programming/Recent content in Integer Programming on Shai AsherHugoen-gbSat, 10 Oct 2026 08:00:00 +0300https://slow-is-smooth.io/blog/the-cleaning-robot-puzzle-the-shortest-route-exactly/Sat, 10 Oct 2026 08:00:00 +0300https://slow-is-smooth.io/blog/the-cleaning-robot-puzzle-the-shortest-route-exactly/<p><em>Part 3&rsquo;s fast route cleans a furnished 20×20 room in 243 commands. The shortest route takes 221, and this part proves that no route takes fewer.</em></p> <p>This is <strong>Part 9</strong> of the cleaning robot puzzle. <a href="https://slow-is-smooth.io/blog/the-cleaning-robot-puzzle-a-lower-bound-from-a-linear-program/">Part 8</a> wrote Part 3&rsquo;s fewest-commands question as a linear program and got a lower bound from it: on that room, 220. The fast route had 243. Somewhere between the two lay the answer, and Part 8 couldn&rsquo;t say where, because it had dropped two rules to make the program linear: that the robot walks between two fields a whole number of times, and that it enters every field as often as it leaves.</p>