在 $\widetilde{O}(n^4)$ 时间内计算最短偶有向环长度
Fast Odd-Permutation Sums in Characteristic Two and Shortest Even Directed Cycles
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中文总结 AI 辅助
该研究提出一种在特征二域上直接评估奇偶环覆盖枚举器的方法，结合逆基闭式与动态逆维护，实现了 $\widetilde{O}(n^4)$ 时间内计算有向图最短偶环长度的蒙特卡洛算法，错误概率为 $O(n^{-3})$。
中文摘要 AI 辅助
Björklund、Husfeldt 和 Kaski 给出了一个随机化的 $\widetilde{O}(n^{\omega+3})$ 时间算法，用于计算 $n$ 顶点有向图中最短偶环的长度，其中 $\omega$ 是方阵乘法指数。他们的代数框架通过将计算提升到特征为四的环上，在特征为二的域上评估奇偶环覆盖枚举器。我们给出了一个在特征二下直接评估非奇异矩阵的方法。如果 $A$ 在特征为二的域上非奇异，那么由奇置换索引的单项式之和可以用 $O(n^3)$ 次域运算确定性计算。关键在于一个基于逆的闭式表达式，用于处理具有两行成比例的矩阵，并结合行消元和动态逆维护。对于有向图，我们将评估器应用于 $I+zW$，其中 $W$ 是随机加权的邻接矩阵。行列式 $\det(I+zW)$ 的常数项系数为 1，因此在任意 $2n+1$ 个不同的域元素中，至少有 $n+1$ 个产生非奇异矩阵。在这些点上进行评估并插值，得到一个蒙特卡洛算法，该算法在 $\widetilde{O}(n^4)$ 时间内计算最短偶环长度，错误概率为 $O(n^{-3})$。该错误关于存在性是一侧的：在没有偶环的图上，算法总是报告该事实。结果涉及长度；标准的自归约在 $\widetilde{O}(n^5)$ 时间内产生一个环。
英文摘要
For a matrix $A$ over a field of characteristic two, let $Φ(A)$ be the sum of its permutation monomials indexed by odd permutations. Although determinant and permanent coincide in this characteristic, this parity sub-sum retains information that neither gives separately. We show that $Φ(A)$ and all its first partial derivatives can be computed deterministically in $O(n^τ)$ field operations for every $n\times n$ matrix, where $2<τ\le3$ is any fixed admissible matrix multiplication exponent. The result includes singular matrices and the binary field. An inversion-count identity expresses $Φ$ through complementary minors. For invertible matrices, a decomposition along two binary interval trees aggregates these minors by matrix multiplication; a Boolean border of constant size handles the remaining ranks. We also give an explicit matrix formula for the full gradient. Applied to $Φ(I+zW)$ for a randomly weighted adjacency matrix $W$, the evaluator computes the shortest even directed-cycle length in $\widetilde{O}(n^{τ+1})$ bit operations, improving the $\widetilde{O}(n^{τ+3})$ bound of Björklund, Husfeldt, and Kaski with the same multiplication exponent. The same time bound recovers the union of the arcs of all shortest even cycles with high probability, and deterministically outputs the cycle under a unique-shortest-cycle promise. For general graphs, exact maintenance of a nonzero coefficient gives an $\widetilde{O}(n^4)$ algorithm that outputs a shortest even cycle with high probability. Complementary extraction methods improve this bound for short cycles and for cycles that omit few vertices.
发表机构
- Peking University（北京大学）
机构由 AI 辅助整理，请以论文原文为准。
