On the Parameterized Intractability of Determinant Maximization
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In the Determinant Maximization problem, given an n× n positive semi-definite matrix A in ℚ^n× n and an integer k, we are required to find a k× k principal submatrix of 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× 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√(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 ϵ-additive approximation algorithm that runs in ϵ^-r^2· r^O(r^3)· n^O(1) time for the rank r of an input matrix, provided that the diagonal entries are bounded.
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