Morse匹配的参数化不可逼近性
Parameterized inapproximability of Morse matching
摘要：
从不可逼近性和参数化复杂性的角度研究了极小化临界单形个数的问题。我们首先证明了Min-Morse匹配在$2^{\log^{(1-ε)}n}$因子内的不可逼近性。我们的第二个结果表明，Min-Morse匹配是${\bf W{[P]}}$ -硬的标准参数。接下来，我们表明，标准参数化的最小莫尔斯匹配没有FPT近似算法的任何近似因子$ρ$。上述硬度结果适用于尺寸为$\ge 2$的配合物。在积极的一面，我们提供了一个因子$O(\frac{n}{\log n})$近似算法的最小莫尔斯匹配$2$ -复合物，注意到没有这样的算法是已知的高维复合物。最后，我们设计了离散梯度与非常少的临界单形的典型实例，从一个相当广泛的参数值的Costa-Farber模型的随机复合物。
英文摘要：
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 $2^{\log^{(1-ε)}n}$. 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 $ρ$. 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.
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