We study the problem of minimizing the number of critical simplices from the point of view of inapproximability and parameterized complexity. We first show inapproximability of Min-Morse Matching within a factor of $2^{\log^{(1-\epsilon)}n}$. Our second result shows that Min-Morse Matching is ${\bf W{[P]}}$-hard with respect to the standard parameter. Next, we show that Min-Morse Matching with standard parameterization has no FPT approximation algorithm for any approximation factor $\rho$. The above hardness results are applicable to complexes of dimension $\ge 2$. On the positive side, we provide a factor $O(\frac{n}{\log n})$ approximation algorithm for Min-Morse Matching on $2$-complexes, noting that no such algorithm is known for higher dimensional complexes. Finally, we devise discrete gradients with very few critical simplices for typical instances drawn from a fairly wide range of parameter values of the Costa-Farber model of random complexes.

中文翻译：

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莫尔斯匹配的参数化不可逼近性

我们从不可逼近性和参数化复杂性的角度研究了最小化临界单纯形数量的问题。我们首先展示了最小莫尔斯匹配在 $2^{\log^{(1-\epsilon)}n}$ 因子内的不可逼近性。我们的第二个结果表明，对于标准参数，Min-Morse 匹配是 ${\bf W{[P]}}$-hard。接下来，我们展示了具有标准参数化的 Min-Morse 匹配对于任何近似因子 $\rho$ 都没有 FPT 近似算法。上述硬度结果适用于维数为$\ge 2$ 的配合物。从积极的方面来说，我们提供了一个因子 $O(\frac{n}{\log n})$ 近似算法用于 $2$-complexes 上的 Min-Morse 匹配，注意到没有这样的算法对于更高维的复数是已知的。最后，