排列增益图上的上下文分数：精确算法、查询下界和动态维护
Contextual Fraction on Permutation Gain Graphs: Exact Algorithms, Query Lower Bounds, and Dynamic Maintenance
AI总结：
研究排列增益图上上下文分数相关问题，通过识别特定排列传输类将全局问题归结为定点计算，给出算法复杂度及查询下界等，证明对某些二元约束语言支持阈值与有限域约束满足问题等价，找到查询最优且可动态维护的易处理岛。
AI中文摘要：
对于显式表示的有限经验模型，判定上下文分数是否严格低于1是NP完全问题，而标准精确线性规划对每个全局赋值都有一列。我们识别出一个排列 - 传输类，其中这个全局问题可归结为定点计算。设连通排列增益图作用于有限状态集\(O\)，\(H\leq{\rm Sym}(O)\)是其完整群子群，\(F = {\rm Fix}(H)\)，\(p\)是\(H\)不变根分布。对于诱导经验模型，\({\rm NCF}(e)=p(F)\)，\({\rm CF}(e)=1 - p(F)\)。因此，兼容性、\(F\)和\({\rm CF}(e)\)可在\(O(|O|(|V| + |E|))\)算术和表格操作中计算。对于每个有限简单2 - 边连通图，在显式排列表查询模型中，任何确定性精确算法在最坏情况下至少需要\((|O| - 1)|E|\)次探测，使对输入表的依赖在常数因子范围内最优。有固定生成树时，弦插入和删除在最坏情况下需要\(O(|O|)\)时间，或与移动集表示成比例的时间，而兼容性和上下文分数查询需要\(O(1)\)时间。最后，对于具有共同边际可实现的二元约束语言，支持阈值\({\rm CF} < 1\)与相关有限域约束满足问题在多项式时间内等价，因此继承了布拉托夫 - 朱克二分法。结果在一般上下文分数问题中识别出一个查询最优且可动态维护的易处理岛。
英文摘要：
For an explicitly represented finite empirical model, deciding whether the contextual fraction is strictly below one is NP-complete, while the standard exact linear program has one column for every global assignment. We identify a permutation-transport class in which this global problem collapses to a fixed-point calculation. Let a connected permutation gain graph act on a finite state set $O$, let $H \leq{ \rm Sym}(O)$ be its holonomy subgroup, let $F = {\rm Fix}(H)$, and let $p$ be an $H$-invariant root distribution. For the induced empirical model, \[ {\rm NCF}(e)=p(F),\qquad {\rm CF}(e)=1-p(F). \] Consequently, compatibility, $F$, and ${\rm CF}(e)$ are computable in $O(|O|(|V|+|E|))$ arithmetic and table operations. For every finite simple $2$-edge-connected graph, any deterministic exact algorithm in the explicit permutation-table query model requires at least $(|O|-1)|E|$ probes in the worst case, making the dependence on the input tables optimal up to constant factors. With a fixed spanning tree, chord insertions and deletions require $O(|O|)$ worst-case time, or time proportional to the moved-set representation, while compatibility and contextual-fraction queries take $O(1)$ time. Finally, for common-marginal realizable binary constraint languages, the support threshold ${\rm CF} < 1$ is polynomial-time equivalent to the associated finite-domain constraint-satisfaction problem and therefore inherits the Bulatov--Zhuk dichotomy. The results identify a query-optimal and dynamically maintainable tractability island inside the general contextual-fraction problem.
