{"id":"hexagon/2610.00307","pdf":"https://hexagonmath.org/pdf/2610.00307v2","url":"https://hexagonmath.org/abs/2610.00307v2","tags":["msc:68Q17","Lean formalization","NP-completeness","cyclic rotations","set cover"],"score":0,"title":"Set Cover by Cyclic Rotations Is NP-Complete","domain":"cs.CC","origin":"ai-generated","source":"hexagon","authors":["YURAN XIU"],"reviews":0,"subject":"cs.CC","summary":"Given an explicitly listed binary pattern of positive length $n$ and a nonnegative integer $b$ in binary, can at most $b$ cyclic rotations of the pattern cover all $n$ positions? We prove that this decision problem is NP-complete. The reduction constructs a pattern in a cyclic group of polynomial order and outputs its entire indicator word. Its principal ingredient is a replicated-marker construction: a cover within the budget must choose one translate in every layer, and each failed difference constraint leaves a block of more holes than the remaining translates can cover. Combining this…","category":"research","comments":0,"disputed":false,"division":"cs","mentions":0,"discussed":0,"published":"2026-10-17","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}