In the Determinant Maximization problem, given an $n\times n$ positive semi-definite matrix $\bf{A}$ in $\mathbb{Q}^{n\times n}$ and an integer $k$, we are required to find a $k\times k$ principal submatrix of $\bf{A}$ having the maximum determinant. This problem is known to be NP-hard and further proven to be W[1]-hard with respect to $k$ by Koutis. However, there is still room to explore its parameterized complexity in the restricted case, in the hope of overcoming the general-case parameterized intractability. In this study, we rule out the fixed-parameter tractability of Determinant Maximization even if an input matrix is extremely sparse or low rank, or an approximate solution is acceptable. We first prove that Determinant Maximization is NP-hard and W[1]-hard even if an input matrix is an arrowhead matrix; i.e., the underlying graph formed by nonzero entries is a star, implying that the structural sparsity is not helpful. By contrast, Determinant Maximization is known to be solvable in polynomial time on tridiagonal matrices. Thereafter, we demonstrate the W[1]-hardness with respect to the rank $r$ of an input matrix. Our result is stronger than Koutis' result in the sense that any $k\times k$ principal submatrix is singular whenever $k>r$. We finally give evidence that it is W[1]-hard to approximate Determinant Maximization parameterized by $k$ within a factor of $2^{-c\sqrt{k}}$ for some universal constant $c>0$. Our hardness result is conditional on the Parameterized Inapproximability Hypothesis posed by Lokshtanov, Ramanujan, Saurab, and Zehavi, which asserts that a gap version of Binary Constraint Satisfaction Problem is W[1]-hard. To complement this result, we develop an $\varepsilon$-additive approximation algorithm that runs in $\varepsilon^{-r^2}\cdot r^{O(r^3)}\cdot n^{O(1)}$ time for the rank $r$ of an input matrix, provided that the diagonal entries are bounded.

翻译：在行列式最大化问题中，给定一个$n\times n$的正半定矩阵$\bf{A}\in\mathbb{Q}^{n\times n}$和一个整数$k$，需要找到$\bf{A}$的一个具有最大行列式的$k\times k$主子矩阵。该问题已知为NP难，并且Koutis进一步证明其关于参数$k$是W[1]难的。然而，在受限情形下，仍有空间探索其参数化复杂性，以期克服一般情形下的参数化难解性。在本研究中，我们排除了行列式最大化的固定参数可解性，即使输入矩阵极其稀疏或低秩，或可接受近似解。我们首先证明，即使输入矩阵为箭形矩阵（即非零项构成的底层图是一个星形），行列式最大化仍是NP难和W[1]难的，这意味着结构稀疏性并无帮助。相比之下，行列式最大化在三对角矩阵上是已知可在多项式时间内求解的。随后，我们展示了关于输入矩阵的秩$r$的W[1]难性。我们的结果在以下意义上强于Koutis的结果：每当$k>r$时，任何$k\times k$主子矩阵都是奇异的。最后，我们给出证据表明，对于某个普适常数$c>0$，在参数$k$下以$2^{-c\sqrt{k}}$因子逼近行列式最大化是W[1]难的。我们的难性结果依赖于Lokshtanov、Ramanujan、Saurab和Zehavi提出的参数化不可逼近性假设，该假设断言二元约束满足问题的间隙版本是W[1]难的。为补充这一结果，我们开发了一个$\varepsilon$-加性近似算法，在输入矩阵的秩$r$下，若对角元有界，该算法运行时间为$\varepsilon^{-r^2}\cdot r^{O(r^3)}\cdot n^{O(1)}$。