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Minimum fill-in: inapproximability and almost tight lower bounds. (English) Zbl 1435.68111
Summary: 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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