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The Hamiltonian Cycle Problem and Travelling Salesman Problem are among famous NP-complete problems and has been studied extensively.

I am looking for applications of the HamCycle and TSP. What are some interesting real world problems where the HamCycle and TSP come up?

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    $\begingroup$ Few years ago, it was useful in genome sequencing. $\endgroup$ – Saeed Jul 1 '16 at 22:59
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One application involves stripification of triangle meshes in computer graphics — a Hamiltonian path through the dual graph of the mesh (a graph with a vertex per triangle and an edge when two triangles share an edge) can be a helpful way to organize data and reduce communication costs.

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Abstract: Multi-threshold CMOS (MTCMOS) is currently the most popular methodology in industry for implementing a power gating design, which can effectively reduce the leakage power by turning off inactive circuit domains. However, large peak current may be consumed in a power-gated domain during its sleep-to-active mode transition. As a result, major IC foundries recommend turning on power switches one by one to reduce the peak current during the mode transition, which requires a Hamiltonian-cycle routing to serially connect all the power switches. ...

  • Another """application""" (note the triple quotes :-) is puzzle games ... for example in the game RoundTrip (a.k.a. GrandTour) you must find an Hamiltonian circuit in a grid of points in which some of the edges are given.

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But there are many other puzzles/videogames that are directly inspired by the Hamiltonian circuit/path problem: Inertia, Pearl, Rolling Cube Puzzles, Slither,...

... and the "hardness" of HC makes them addictive: even small instances can be very hard to solve for our brain!!!

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(Variants of the) TSP show(s) up routinely in several routing and scheduling problems (think of the route planned for a UPS truck out of delivering packages, for instance). But a great place to learn about several applications of the TSP is Chapter 3 of the book In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation by Bill Cook. In particular, he talks about mapping genomes; aiming telescopes, X-rays, and lasers; guiding industrial machines; organizing data; tests for microprocessors, job scheduling, and more.

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The TSP often appears as part of more "complicated" operations research problems. A good example being the vehicle routing problem which has many variants. The following web site tracks some of the developments in this area:

http://neo.lcc.uma.es/vrp/vehicle-routing-problem/

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