>>12356675Ignoring the shitposting going on this thread, the HamCycle problem is incredibly fundamental in areas that are heavy for research into optimization. So anything having to do with:
>The Travelling Salesman Problem and simulated annealingEverything from doing tests on microprocessors to aiming telescopes to vehicular routing works with interesting instances of TSP. Here's an application of Hamiltonian Cycles to CMOS design:
https://sci-hub.do/10.1109/aspdac.2012.6165026If you want a more theoretically charged answer, Hamiltonian Cycles are part of an important class of NP-complete problems that can be identified to be similar in flavor - this lets us understand why some approaches don't work for *classes* of problems rather than individual problems, even if they're all NP-complete. For example, there are a multitude of barriers to providing an answer to P vs. NP, and we've basically figured out that separating the difficulty of the permanent vs. the determinant is the *easiest* approach, and even this approach is only in its infancy. There's a good post on stackexchange relating HamCycle and the Clique problem using monotone projections:
https://cstheory.stackexchange.com/questions/27496/why-is-hamiltonian-cycle-so-different-from-permanent?rq=1There are likely many more reasons, but these are the ones that were on the top of my head
