On the Parameterized Intractability of Determinant Maximization

TL;DR

This paper investigates the parameterized computational complexity of the Determinant Maximization problem, proving its intractability even under restrictive conditions and establishing hardness of approximation, while also providing an approximation algorithm.

Contribution

It demonstrates the W[1]-hardness of Determinant Maximization for sparse, low-rank, and approximate cases, and introduces an approximation algorithm for matrices with bounded diagonal entries.

Findings

Determinant Maximization is NP-hard and W[1]-hard for arrowhead matrices.

The problem remains W[1]-hard when parameterized by the matrix rank.

It is W[1]-hard to approximate within a factor of 2^{-c√k} under certain hypotheses.

Abstract

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.