The concept of NP-completeness has been proposed for half a century, and it is conjectured that there are no subexponential-time algorithms for NP-hard problems, which is known as the Exponential Time Hypothesis (ETH). As a pivotal conjecture in the field of theoretical computer science, numerous conjectures in computer science rely on ETH. A corollary of the Exponential Time Hypothesis is the Counting Exponential Time Hypothesis ($\#ETH$), and a further corollary of $\#ETH$ is that $\#W[1] \neq \text{FPT}$. The $\#k$-matching problem is a well-known $\#W[1]$-complete problem. We have discovered an algorithm for the $\#k$-matching problem with a running time of $f(k)n^{O(1)}$. This result implies that the hypotheses $\#W[1] \neq \text{FPT}$, $W[1] \neq \text{FPT}$, the Counting Exponential Time Hypothesis, and the Exponential Time Hypothesis all do not hold.

翻译：NP完全性的概念已被提出半个世纪，人们推测NP难问题不存在亚指数时间算法，这被称为指数时间假说（ETH）。作为理论计算机科学领域的关键猜想，计算机科学中的众多猜想都依赖于ETH。指数时间假说的一个推论是计数指数时间假说（$\#ETH$），而$\#ETH$的进一步推论是$\#W[1] \neq \text{FPT}$。$\#k$-匹配问题是一个著名的$\#W[1]$-完全问题。我们发现了一种针对$\#k$-匹配问题的算法，其运行时间为$f(k)n^{O(1)}$。这一结果表明，$\#W[1] \neq \text{FPT}$、$W[1] \neq \text{FPT}$、计数指数时间假说以及指数时间假说均不成立。