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

Abstract

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.