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Complete Solution of Tropical Vector Inequalities Using Matrix Sparsification

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Abstract

We examine the problem of finding all solutions of two-sided vector inequalities given in the tropical algebra setting, where the unknown vector multiplied by known matrices appears on both sides of the inequality. We offer a solution that uses sparse matrices to simplify the problem and to construct a family of solution sets, each defined by a sparse matrix obtained from one of the given matrices by setting some of its entries to zero. All solutions are then combined to present the result in a parametric form in terms of a matrix whose columns form a complete system of generators for the solution. We describe the computational technique proposed to solve the problem, remark on its computational complexity and illustrate this technique with numerical examples.

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Acknowledgments

The author sincerely thanks the anonymous referee for the insightful comments, valuable suggestions and corrections, which have been in-corporated in the revised paper. He is especially grateful for providing a list of important references and for giving a simple solution for Example 6.1 from scratch.

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Authors and Affiliations

  1. St. Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 199034, Russia

    Nikolai Krivulin

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  1. Nikolai Krivulin
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Correspondence to Nikolai Krivulin.

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This work was supported in part by the Russian Foundation for Basic Research under the grant 20-010-00145.

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Krivulin, N. Complete Solution of Tropical Vector Inequalities Using Matrix Sparsification. Appl Math 65, 755–775 (2020). https://doi.org/10.21136/AM.2020.0376-19

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