从一个解到多个解：基于神示元定理的通用多样性测度下多样化解的FPT框架
From One Solution to Many: An Oracle-Based FPT Framework for Diverse Solutions under Generalized Diversity Measures
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中文总结 AI 辅助
该研究针对通用多样性测度下的多样化解问题，提出基于神示的FPT框架，改进了神示调用复杂度，统一并增强了现有方法，为相关经典问题提供了更优的固定参数可处理算法。
中文摘要 AI 辅助
计算多样化解的问题近年来已成为重要研究领域，其研究动机来自公平性、鲁棒性和安全性等应用场景。该问题的目标不是返回单个可行或最优解，而是输出一组具有显著差异的解，通常通过对称差来衡量差异程度。多样化解变体已通过稀疏化、网络流归约和代数技术等方法进行研究。我们研究了隐式集合系统问题的多样化解变体的固定参数可处理性：给定参数k、r和阈值b，任务是计算r个可行解，每个解的大小至多为k，且在指定目标下的多样性至少为b。我们的主要贡献是一个基于神示元定理，我们确定了一类广泛的目标，称为“一致多样”，其中包含若干标准测度。假设存在一个“精确空扩展神示”，给定禁止集Forb，该神示返回指定大小的可行解，且该解避开Forb，或报告不存在这样的解；我们获得了一个以k+r为参数的固定参数可处理算法，该算法最多进行(2kr)^kr · r次神示调用，且每次调用中神示参数满足s+|Forb| ≤ k+2kr。我们的框架统一并增强了之前基于神示的方法：与Kumabe在ESA 2025提出的框架（其神示调用次数为双指数界）相比，我们的方法实现了单指数界2^{O(kr log(kr))}，并直接构造所需的解元组。我们恢复了该框架覆盖的所有问题的固定参数可处理算法，且神示复杂度得到改进，同时为经典图和拟阵问题的多样化解变体获得了强界。
英文摘要
The problem of computing \emph{diverse} solutions has recently emerged as an important area of study, motivated by applications in fairness, robustness, and security. Instead of returning a single feasible or optimal solution, the goal is to output a \emph{collection} of meaningfully different solutions, often measured by symmetric differences. Diverse variants have been studied using sparsification, network-flow reductions, and algebraic techniques. We investigate the fixed-parameter tractability of diverse variants of an implicit set-system problem. Given parameters $k$ and $r$ and a threshold $b$, the task is to compute $r$ feasible solutions, each of size at most $k$, whose diversity under a specified objective is at least $b$. Our main contribution is an oracle-based meta-theorem. We identify a broad class of objectives, called \emph{consistently diverse}, that includes several standard measures. Assuming an \emph{exact empty-extension oracle} given a forbidden set ${\sf Forb}$, which returns a feasible solution of a prescribed size avoiding ${\sf Forb}$ or reports that none exists, we obtain a fixed-parameter tractable algorithm parameterized by $k+r$. The algorithm makes at most $(2kr)^{kr} \cdot r$ oracle calls, and in each call the oracle parameter satisfies $s+|{\sf Forb}| \leq k+2kr$. Our framework unifies and strengthens previous oracle-based approaches. Compared with Kumabe's framework (ESA 2025), which gives a doubly exponential bound on the number of oracle calls, our approach achieves the single exponential bound $2^{O(kr\log(kr))}$ and directly constructs the desired tuple of solutions. We recover fixed-parameter tractable algorithms for all problems covered by that framework, with improved oracle complexity, and obtain strong bounds for diverse variants of classical graph and matroid problems.
