在有界团宽图上对最优MSO可定义集进行量化
Extending Courcelle's Theorem with Optimality Predicates
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中文总结 AI 辅助
该研究引入$\boldsymbol{\textsf{AmCMSO}}$逻辑，针对有界团宽、树宽图建立其模型检测的固定参数可处理元定理，解决了双层图优化等问题，同时证明了依赖外部集合变量的最优性谓词会使模型检测变难。
中文摘要 AI 辅助
我们引入了$\boldsymbol{\textsf{AmCMSO}}$，它是计数型一元二阶逻辑（$\boldsymbol{\textsf{CMSO}}$）的扩展，新增了可引用最小和最大值满足赋值的谓词。我们针对有界团宽图建立了$\boldsymbol{\textsf{AmCMSO}}_1$的固定参数可处理模型检测元定理，以及针对有界树宽图建立了$\boldsymbol{\textsf{AmCMSO}}_2$的同类元定理。这些元定理为多个双层图优化问题提供了固定参数可处理算法，包括用于解唯一化的阻断和预分配问题，以及无需以最优值为参数即可最大化最优解多样性的算法。相反，若允许最优性谓词依赖外部集合变量，即使在固定深度的树上，模型检测也会对多项式分层的每一层都变得困难。
英文摘要
Courcelle's theorem and its optimization variants yield fixed-parameter tractable algorithms for a wide range of graph problems on graphs of bounded treewidth or clique-width. However, the limited counting power of $\mathsf{CMSO}$ poses an obstacle to capturing certain optimization problems and properties within this framework. We introduce a new logic $\mathsf{AmCMSO}$, which extends $\mathsf{CMSO}$ with predicates for membership in the families of minimum- and maximum-cardinality sets satisfying a fixed formula $ϕ(X)$. In contrast to most previous extensions of $\mathsf{CMSO}$ with cardinality constraints, we give algorithmic meta-theorems based on fixed-parameter tractable model checking for $\mathsf{AmCMSO}_1$ parameterized by clique-width and the formula, and for $\mathsf{AmCMSO}_2$ parameterized by treewidth and the formula. Our proof is based on the combination of Feferman--Vaught-type decomposition and fundamental techniques for dynamic programming. The meta-theorems yield fixed-parameter tractable algorithms for a wide range of optimization problems involving optimal solutions, including network interdiction, pre-assignment for solution uniquification, and diversity maximization, without parameterizing by the optimum value. Finally, allowing an optimality predicate to depend on even one external set variable makes model checking hard for every level of the polynomial hierarchy, already on trees of depth four.
发表机构
- Hokkaido University（北海道大学）
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