Reachability, distance, and matching are some of the most fundamental graph problems that have been of particular interest in dynamic complexity theory in recent years [DKMSZ18, DMVZ18, DKMTVZ20]. Reachability can be maintained with first-order update formulas, or equivalently in DynFO in general graphs with n nodes [DKMSZ18], even under O(log n/loglog n) changes per step [DMVZ18]. In the context of how large the number of changes can be handled, it has recently been shown [DKMTVZ20] that under a polylogarithmic number of changes, reachability is in DynFOpar in planar, bounded treewidth, and related graph classes -- in fact in any graph where small non-zero circulation weights can be computed in NC. We continue this line of investigation and extend the meta-theorem for reachability to distance and bipartite maximum matching with the same bounds. These are amongst the most general classes of graphs known where we can maintain these problems deterministically without using a majority quantifier and even maintain witnesses. For the bipartite matching result, modifying the approach from [FGT], we convert the static non-zero circulation weights to dynamic matching-isolating weights. While reachability is in DynFOar under O(log n/loglog n) changes, no such bound is known for either distance or matching in any non-trivial class of graphs under non-constant changes. We show that, in the same classes of graphs as before, bipartite maximum matching is in DynFOar under O(log n/loglog n) changes per step. En route to showing this we prove that the rank of a matrix can be maintained in DynFOar, also under O(log n/loglog n) entry changes, improving upon the previous O(1) bound [DKMSZ18]. This implies similar extension for the non-uniform DynFO bound for maximum matching in general graphs and an alternate algorithm for maintaining reachability under O(log n/loglog n) changes [DMVZ18].

中文翻译：

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距离和匹配的动态元定理

可达性、距离和匹配是近年来动态复杂性理论特别感兴趣的一些最基本的图问题 [DKMSZ18、DMVZ18、DKMTVZ20]。可达性可以通过一阶更新公式来维护，或者等效地在 DynFO 中在具有 n 个节点的一般图中 [DKMSZ18]，即使在每步 [DMVZ18] 的 O(log n/loglog n) 变化下也是如此。在可以处理的变化数量有多大的背景下，最近已经表明 [DKMTVZ20] 在多对数变化下，可达性在平面、有界树宽和相关图类中的 DynFOpar 中——实际上在任何可以在 NC 中计算小的非零循环权重的图。我们继续这一研究路线，并将可达性元定理扩展到距离和具有相同边界的二分最大匹配。这些是已知的最通用的图类之一，我们可以在其中确定性地维护这些问题，而无需使用多数量词，甚至可以维护证人。对于二分匹配结果，修改 [FGT] 中的方法，我们将静态非零循环权重转换为动态匹配隔离权重。虽然可达性在 O(log n/loglog n) 变化下的 DynFOar 中，但在非常规变化下的任何非平凡类图中，距离或匹配都没有这样的界限。我们表明，在与以前相同的图类中，二部最大匹配在 DynFOar 中在每步 O(log n/loglog n) 变化下进行。在展示这一点的过程中，我们证明矩阵的秩可以在 DynFOar 中保持，也在 O(log n/loglog n) 条目更改下，改进了之前的 O(1) 界限 [DKMSZ18]。