Given an $n\times n$ sparse symmetric matrix with m nonzero entries, performing Gaussian elimination may turn some zeroes into nonzero values, so called fill-ins. The minimum fill-in problem asks whether it is possible to perform the elimination with at most k fill-ins. We exclude the existence of polynomial time approximation schemes for this problem, assuming P≠NP, and the existence of $2^{O(n^{1-\delta })}$-time approximation schemes for any positive δ, assuming the Exponential Time Hypothesis. We also give a $2^{O(k^{1/2-\delta })}\cdot n^{O(1)}$ parameterized lower bound. All these results come as corollaries of a new reduction from vertex cover to the minimum fill-in problem, which might be of its own interest: All previous reductions for similar problems start from some kind of graph layout problems, and hence have limited use in understanding their fine-grained complexity.

给定一个 $ñ\times ñ$具有m个非零条目的稀疏对称矩阵，执行高斯消除可能会将一些零变成非零值，因此称为填充。最小填充问题询问是否有可能执行最多k个填充的消除。假设P≠NP，并且存在P的存在，我们排除了针对该问题的多项式时间逼近方案。$2^{Ø（{ñ}^{\mathrm{1个}-\delta }）}$假设指数时间假设，任何正δ的时间近似方案。我们还给$2^{Ø（{ķ}^{\mathrm{1个}/2-\delta }）}\cdot {ñ}^{Ø（\mathrm{1个}）}$参数化的下界。所有这些结果都是从顶点覆盖到最小填充问题的新减少的推论得出的，这可能与它自己的兴趣有关：以前对类似问题的所有减少都从某种图形布局问题开始，因此在了解其细粒度的复杂性。