Matrix Rigidity from the Viewpoint of Parameterized Complexity

Fedor V. Fomin, Daniel Lokshtanov, S. M. Meesum, Saket Saurabh, Meirav Zehavi
2018 SIAM Journal on Discrete Mathematics  
The rigidity of a matrix A for a target rank r over a field F is the minimum Hamming distance between A and a matrix of rank at most r. Rigidity is a classical concept in Computational Complexity Theory: constructions of rigid matrices are known to imply lower bounds of significant importance relating to arithmetic circuits. Yet, from the viewpoint of Parameterized Complexity, the study of central properties of matrices in general, and of the rigidity of a matrix in particular, has been
more » ... d. In this paper, we conduct a comprehensive study of different aspects of the computation of the rigidity of general matrices in the framework of Parameterized Complexity. Naturally, given parameters r and k, the Matrix Rigidity problem asks whether the rigidity of A for the target rank r is at most k. We show that in case F = R or F is any finite field, this problem is fixed-parameter tractable with respect to k + r. To this end, we present a dimension reduction procedure, which may be a valuable primitive in future studies of problems of this nature. We also employ central tools in Real Algebraic Geometry, which are not well known in Parameterized Complexity, as a black box. In particular, we view the output of our dimension reduction procedure as an algebraic variety. Our main results are complemented by a W[1]-hardness result and a subexponential-time parameterized algorithm for a special case of Matrix Rigidity, highlighting the different flavors of this problem. special case of adjacency matrices, and therefore they are primarily studies in Graph Theory rather than Matrix Theory [16, 17] . In this paper, we conduct a comprehensive study of different aspects of the computation of the rigidity of general matrices in the framework of Parameterized Complexity. Formally, given a matrix A over a field F, the rigidity of A, denoted by R F A (r), is defined as the minimum Hamming distance between A and a matrix of rank at most r. In other words, R F A (r) is the minimum number of entries in A that need to be edited in order to obtain a matrix of rank at most r. Naturally, given parameters r and k, the Matrix Rigidity problem asks whether R F A (r) ≤ k. Input: A matrix A with each entry an integer, and two non-negative integers r, k. Question: Is R R A (r) ≤ k? FF Matrix Rigidity Parameter: p, r, k 1 Editing an edge {u, v} means that if {u, v} ∈ E then {u, v} is deleted, and otherwise it is added. F. V. Fomin, D. Lokshtanov, S. M. Meesum, S. Saurabh, and M. Zehavi 32:3 in time O * (2 O(f (r) √ k log k) ). Meesum and Saurabh [17] obtained similar results for directed graphs. Lemma 8. Let A be a matrix over some field F, and let r and k be two non-negative integers. Given an instance (A, r, k), the procedure Matrix-Reduction runs in time polynomial in input size and returns a matrix A satisfying the following properties: 1. A has at most O(r 2 · k 2 ) entries. If the output is produced by lines 6c and 9 of Column-Reduction (when called by Matrix-Reduction), then A is a jumbled submatrix of A. (A, r, k) is a YES-instance of Matrix Rigidity if and only if ( A, r, k) is a YES-instance. Proof. The steps of procedure Column-Reduction are all computable in polynomial time, and therefore Matrix-Reduction runs in polynomial time. We now prove the desired properties one by one. Let the matrix N denote the output of Column-Reduction on the input instance (N, r, k) . Proof of 1. We first bound the size of the output of Column-Reduction. The output of this procedure can occur at lines 1, 4, 6c and 9. If the output happens at line 1, it has 1 column S TA C S 2 0 1 7 S TA C S 2 0 1 7
doi:10.1137/17m112258x fatcat:wbhtcizaxna3vegoyshtwxwzlu