Quantifying over Optimal MSO-Definable Sets on Graphs of Bounded Clique-Width

2026-08-20

AI Summary

研究通过扩展计数单点序逻辑并引入最优解谓词，为固定团宽和树宽图上的双层图优化问题提供了固定参数可处理的算法。

Abstract

We introduce $\mathsf{AmCMSO}$, an extension of counting monadic second-order logic ($\mathsf{CMSO}$) with predicates that refer to minimum- and maximum-value satisfying assignments. We establish fixed-parameter tractable model-checking meta-theorems for $\mathsf{AmCMSO}_1$ on graphs of bounded clique-width and for $\mathsf{AmCMSO}_2$ on graphs of bounded treewidth. These meta-theorems yield fixed-parameter tractable algorithms for several bilevel graph optimization problems, including interdiction and preassignment problems for solution uniquification, as well as algorithms for maximizing the diversity of optimal solutions without parameterizing by the optimum value. In contrast, allowing an optimality predicate to depend on an external set variable makes model checking hard for every level of the polynomial hierarchy, even on trees of fixed depth.

Problem

• Clique-Width
• Model-Checking
• Optimization Problems

Innovation

• AmCMSO
• fixed-parameter tractable model-checking
• bilevel graph optimization
• clique-width
• treewidth