$\mathrm{\#}$W[1] = $\text{FPT}$: Fixed-Parameter Tractable Exact Algorithms for the $\mathrm{\#}k$-Matching Problem

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 ($\mathrm{\#}ETH$), and a further corollary of $\mathrm{\#}ETH$ is that $\mathrm{\#}W[1]\mathrm{\ne }\text{FPT}$. The $\mathrm{\#}k$-matching problem is a well-known $\mathrm{\#}W[1]$-complete problem. We have discovered an algorithm for the $\mathrm{\#}k$-matching problem with a running time of $f(k)n^{O(1)}$. This result implies that the hypotheses $\mathrm{\#}W[1]\mathrm{\ne }\text{FPT}$, $W[1]\mathrm{\ne }\text{FPT}$, the Counting Exponential Time Hypothesis, and the Exponential Time Hypothesis all do not hold.