Actually, on second thought, this isn't quite so easily isomorphic to traveling salesman (any particular "order shredded strips" problem is easily equivalent to

Actually, on second thought, this isn't quite so easily isomorphic to traveling salesman (any particular "order shredded strips" problem is easily equivalent to a traveling salesman problem, but I'm not positive that an arbitrary traveling salesman problem can be re-cast as an "order shredded strips" problem in a consistent manner, so I shouldn't make that claim). So I retract my statement, it's quite possible that someone could effectively solve this problem for reasonable inputs without quadratic runtime (though I'm not positive exactly how you would do it, and the efficacy of your method would likely depend on your assumptions about the inputs).