An Extension of Plücker Relations with Applications to Subdeterminant Maximization
Given a matrix A and k ≥ 0, we study the problem of finding the k × k submatrix of A with the maximum determinant in absolute value. This problem is motivated by the question of computing the determinant-based lower bound of cite{LSV86} on hereditary discrepancy, which was later shown to be an approximate upper bound as well [Matoušek, 2013]. The special case where k coincides with one of the dimensions of A has been extensively studied. Nikolov gave a 2^{O(k)}-approximation algorithm for this special case, matching known lower bounds; he also raised as an open problem the question of designing approximation algorithms for the general case.
We make progress towards answering this question by giving the first efficient approximation algorithm for general k× k subdeterminant maximization with an approximation ratio that depends only on k. Our algorithm finds a k^{O(k)}-approximate solution by performing a simple local search. Our main technical contribution, enabling the analysis of the approximation ratio, is an extension of Plücker relations for the Grassmannian, which may be of independent interest; Plücker relations are quadratic polynomial equations involving the set of k× k subdeterminants of a k× n matrix. We find an extension of these relations to k× k subdeterminants of general m× n matrices.
Plücker relations
determinant maximization
local search
exchange property
discrete concavity
discrepancy
Theory of computation~Approximation algorithms analysis
Theory of computation~Randomness, geometry and discrete structures
Theory of computation~Randomized local search
56:1-56:16
APPROX
A full version of the paper is available at https://arxiv.org/abs/2004.13018.
We would like to thank Aleksandar Nikolov for initial discussions about general subdeterminant maximization.
Nima
Anari
Nima Anari
Department of Computer Science, Stanford University, CA, USA
Thuy-Duong
Vuong
Thuy-Duong Vuong
Department of Computer Science, Stanford University, CA, USA
10.4230/LIPIcs.APPROX/RANDOM.2020.56
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Nima Anari and Thuy-Duong Vuong
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