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arXiv · 2401.14963

On the Hardness of Gray Code Problems for Combinatorial Objects

Arturo Merino·Namrata·Aaron Williams
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Abstract

Can a list of binary strings be ordered so that consecutive strings differ in a single bit? Can a list of permutations be ordered so that consecutive permutations differ by a swap? Can a list of non-crossing set partitions be ordered so that consecutive partitions differ by refinement? These are examples of Gray coding problems: Can a list of combinatorial objects (of a particular type and size) be ordered so that consecutive objects differ by a flip (of a particular type)? For example, 000, 001, 010, 100 is a no instance of the first question, while 1234, 1324, 1243 is a yes instance of the second question due to the order 1243, 1234, 1324. We prove that a variety of Gray coding problems are NP-complete using a new tool we call a Gray code reduction.

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Arturo Merino, Namrata, Aaron Williams. 2024-01-26. On the Hardness of Gray Code Problems for Combinatorial Objects. https://arxiv.org/abs/2401.14963

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