有界树深度图上的严格NP问题的统一算法

中文摘要 AI 辅助

一个经典的良拟序结果保证了在有界树深度的关系结构上，严格NP中的所有问题都存在非均匀线性时间算法；然而，这既没有提供构造这些算法的过程，也没有提供其参数依赖性的可计算界限。我们将这一存在性结果转化为一个统一的算法元定理。给定一个严格NP语句$\varphi$和一个关系结构$\mathcal{R}$，我们的算法在时间$f(|\varphi|, td(\mathcal{R})) \cdot |\mathcal{R}|$内判定$\mathcal{R}\models\varphi$是否成立，其中$f$是一个可计算函数，$\mathcal{R}$的树深度在Gaifman图上度量。该算法还构造见证关系，其多项式指数取决于它们的元数，并为解决以树深度为参数化的遗传图问题提供了一个统一框架。我们还给出了几个应用——其中，我们的结果解决了关于计算栈数、队列数、轨道数和孪生宽度（以树深度为参数）的固定参数可处理性的开放问题。

英文摘要

A classical well-quasi-ordering result guarantees the existence of non-uniform linear-time algorithms for all problems in Strict NP on relational structures of bounded treedepth; however, this provides neither a procedure for constructing these algorithms nor computable bounds on their parameter dependence. We turn this existential result into a uniform algorithmic metatheorem. Given a Strict NP sentence $φ$ and a relational structure $\mathcal{R}$, our algorithm decides whether $\mathcal{R}\modelsφ$ in time $f(|φ|, td(\mathcal{R})) \cdot |\mathcal{R}|$ for a computable function $f$, where the treedepth of $\mathcal{R}$ is measured on the Gaifman graph. The algorithm also constructs witness relations, with the polynomial exponent depending on their arity, and provides a unified framework for settling hereditary graph problems parameterized by treedepth. We also present several applications - among others, our result resolves open questions on the fixed-parameter tractability of computing the stack number, queue number, track number and twin-width parameterized by treedepth.

发表机构

• TU Wien（维也纳工业大学）
• CISPA Helmholtz Center for Information Security and Saarland University（CISPA赫尔姆霍兹信息安全中心与萨尔兰大学）

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