扩展编码
Extensor-Coding
摘要：
我们设计了一个算法，近似计算的路径长度$k$在一个给定的有向图的$n$顶点的数量高达一个乘法误差$1 \pm \varepsilon$。我们的算法运行在时间$\varepsilon^{-2} 4^k(n+m) \operatorname{poly}(k)$。该算法是基于与每个顶点关联的外部（或格拉斯曼）代数中的元素，称为扩展器，然后在此代数中执行计算。这种连接到外部代数推广了以前的最长路径问题的一些方法，是独立的概念感兴趣。使用这种方法，我们也得到了一个确定性的$2^{k}\cdot\operatorname{poly}(n)$时间算法，找到一个$k$ -路径在一个给定的有向图，承诺有几个。我们的结果和技巧推广到子图同构问题时，我们正在寻找的子图有界路宽。最后，我们还得到了一个随机算法来检测$k$ -多线性项的多元多项式作为一个一般的代数电路。据我们所知，这以前只知道代数电路不涉及负常数。
英文摘要：
We devise an algorithm that approximately computes the number of paths of length $k$ in a given directed graph with $n$ vertices up to a multiplicative error of $1 \pm \varepsilon$. Our algorithm runs in time $\varepsilon^{-2} 4^k(n+m) \operatorname{poly}(k)$. The algorithm is based on associating with each vertex an element in the exterior (or, Grassmann) algebra, called an extensor, and then performing computations in this algebra. This connection to exterior algebra generalizes a number of previous approaches for the longest path problem and is of independent conceptual interest. Using this approach, we also obtain a deterministic $2^{k}\cdot\operatorname{poly}(n)$ time algorithm to find a $k$-path in a given directed graph that is promised to have few of them. Our results and techniques generalize to the subgraph isomorphism problem when the subgraphs we are looking for have bounded pathwidth. Finally, we also obtain a randomized algorithm to detect $k$-multilinear terms in a multivariate polynomial given as a general algebraic circuit. To the best of our knowledge, this was previously only known for algebraic circuits not involving negative constants.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
