On the Parameterized Intractability of Determinant Maximization

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.