Minimum fill-in: inapproximability and almost tight lower bounds.

*(English)*Zbl 1435.68111Summary: 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 \(\mathrm{P} \neq \mathrm{NP}\), and the existence of \(2^{O(n^{1 - \delta})}\)-time approximation schemes for any positive \(\delta\), 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.

##### MSC:

68Q17 | Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) |

05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |

65F50 | Computational methods for sparse matrices |

68R10 | Graph theory (including graph drawing) in computer science |

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\textit{Y. Cao} and \textit{R. B. Sandeep}, Inf. Comput. 271, Article ID 104514, 9 p. (2020; Zbl 1435.68111)

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