On the Hardness of Gray Code Problems for Combinatorial Objects

TL;DR

This paper proves that many Gray coding problems for combinatorial objects are NP-complete by introducing a new reduction technique, highlighting their computational hardness.

Contribution

The paper introduces Gray code reduction, a novel method to establish NP-completeness of various Gray coding problems for combinatorial objects.

Findings

Several Gray code problems are NP-complete.

Gray code reduction effectively proves computational hardness.

Highlights the complexity of ordering combinatorial objects with minimal differences.

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.