发表机构
Kyushu University（九州大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究带线性侧的二次字方程，证明其 Nielsen 图直径为多项式，从而确立该方程类可满足性的 NP 完全性。
AI 中文摘要
字方程的可满足性问题询问变量是否可以被替换为单词，使得方程两侧相等。对于正则字方程，其中每个变量在每一侧最多出现一次，可满足性是 NP 完全的。对于一般的二次字方程，其中每个变量总共最多出现两次，可满足性是 NP 难的，但其是否属于 NP 仍是开放问题。我们考虑一个中间类：带线性侧的二次字方程，其中每个变量在指定的一侧最多出现一次。我们证明，此类中方程 $U=V$ 的 Nielsen 图，总长度 $N=|U|+|V|$，其直径（在可达顶点对之间测量）为 $O(N^{12})$。结合已知的正则字方程的 NP 难性，这一结果确立了此类方程可满足性的 NP 完全性。
英文摘要
The satisfiability problem for word equations asks whether variables can be replaced by words so that the two sides become equal. For regular word equations, in which each variable occurs at most once on each side, satisfiability is NP-complete. For general quadratic word equations, in which each variable occurs at most twice in total, satisfiability is NP-hard, but its membership in NP remains open. We consider an intermediate class: quadratic word equations with a linear side, where each variable occurs at most once on one designated side. We show that the Nielsen graph of an equation $U=V$ in this class, with total length $N=|U|+|V|$, has diameter $O(N^{12})$, measured over reachable pairs of vertices. Together with the known NP-hardness for regular word equations, this result establishes NP-completeness of satisfiability for this class. We also show that each strongly connected component is isomorphic to the length-preserving reachability graph of a regular equation, and that the condensation graph has depth at most $|U|+2|V|$.
