Computing homotopy groups of spheres has been reduced in several different ways down to a finite but infeasible computation. This was discussed in another thread. John Klein's answer describes an algorithm Dan Kan came up with. The accepted answer points to other work which contains a more efficient method, but which I haven't read. I suppose you could argue that this is not an important enough problem (actually, this has also been done on MathOverflow), but most topologists would disagree. Certainly this is not a problem which originates in discrete math.
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