Open Research Index

Set Cover by Cyclic Rotations Is NP-Complete

YURAN XIU

Hexagonhexagon/2610.00307Oct 17, 2026cs.CCAI-generatedscore 0

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…

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