Applications of Mathematics, first online, pp. 1-21


Complete solution of tropical vector inequalities using matrix sparsification

Nikolai Krivulin

Received December 28, 2019.   Published online September 16, 2020.

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.
Keywords:  tropical semifield; tropical two-sided inequality; matrix sparsification; complete solution; backtracking
Classification MSC:  15A80, 15A39, 65F50
DOI:  10.21136/AM.2020.0376-19

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References:
[1] M. Akian, S. Gaubert, A. Guterman: Tropical polyhedra are equivalent to mean payoff games. Int. J. Algebra Comput. 22 (2012), Article ID 1250001, 43 pages. DOI 10.1142/S0218196711006674 | MR 2900854 | Zbl 1239.14054
[2] X. Allamigeon, S. Gaubert, R. D. Katz: The number of extreme points of tropical polyhedra. J. Comb. Theory, Ser. A 118 (2011), 162-189. DOI 10.1016/j.jcta.2010.04.003 | MR 2737191 | Zbl 1246.14078
[3] F. L. Baccelli, G. Cohen, G. J. Olsder, J.-P. Quadrat: Synchronization and Linearity: An Algebra for Discrete Event Systems. Wiley Series in Probability and Statistics. Wiley, Chichester (1992). MR 1204266 | Zbl 0824.93003
[4] P. Butkovič: Max-Linear Systems: Theory and Algorithms. Springer Monographs in Mathematics. Springer, London (2010). DOI 10.1007/978-1-84996-299-5 | MR 2681232 | Zbl 1202.15032
[5] P. Butkovič, G. Hegedüs: An elimination method for finding all solutions of the system of linear equations over an extremal algebra. Ekon.-Mat. Obz. 20 (1984), 203-215. MR 0782401 | Zbl 0545.90101
[6] P. Butkovič, K. Zimmermann: A strongly polynomial algorithm for solving two-sided linear systems in max-algebra. Discrete Appl. Math. 154 (2006), 437-446. DOI 10.1016/j.dam.2005.09.008 | MR 2203194 | Zbl 1090.68119
[7] B. A. Carré: An algebra for network routing problems. J. Inst. Math. Appl. 7 (1971), 273-294. DOI 10.1093/imamat/7.3.273 | MR 0292583 | Zbl 0219.90020
[8] A. H. Clifford, G. B. Preston: The Algebraic Theory of Semigroups. Vol. 1. Mathematical Surveys and Monographs 7. American Mathematical Society, Providence (1961). DOI 10.1090/surv/007.1 | MR 0132791 | Zbl 0111.03403
[9] R. A. Cuninghame-Green: Projections in minimax algebra. Math. Program. 10 (1976), 111-123. DOI 10.1007/BF01580656 | MR 0403664 | Zbl 0336.90062
[10] R. A. Cuninghame-Green: Minimax Algebra. Lecture Notes in Economics and Mathematical Systems 166. Springer, Berlin (1979). DOI 10.1007/978-3-642-48708-8 | MR 0580321 | Zbl 0399.90052
[11] R. A. Cuninghame-Green: Minimax algebra and applications. Advances in Imaging and Electron Physics. Volume 90. Academic Press, San Diego, 1994, 1-121. DOI 10.1016/S1076-5670(08)70083-1
[12] R. A. Cuninghame-Green, P. Butkovič: The equation $A\otimes x=B\otimes y$ over (max,+). Theor. Comput. Sci. 293 (2003), 3-12. DOI 10.1016/S0304-3975(02)00228-1 | MR 1957609 | Zbl 1021.65022
[13] R. A. Cuninghame-Green, P. Butkovič: Bases in max-algebra. Linear Algebra Appl. 389 (2004), 107-120. DOI 10.1016/j.laa.2004.03.022 | MR 2080398 | Zbl 1059.15001
[14] L. Elsner, P. van den Driessche: Max-algebra and pairwise comparison matrices. II. Linear Algebra Appl. 432 (2010), 927-935. DOI 10.1016/j.laa.2009.10.005 | MR 2577637 | Zbl 1191.15019
[15] S. Gaubert, R. D. Katz: The tropical analogue of polar cones. Linear Algebra Appl. 431 (2009), 608-625. DOI 10.1016/j.laa.2009.03.012 | MR 2535537 | Zbl 1172.52002
[16] S. Gaubert, R. D. Katz, S. Sergeev: Tropical linear-fractional programming and parametric mean payoff games. J. Symb. Comput. 47 (2012), 1447-1478. DOI 10.1016/j.jsc.2011.12.049 | MR 2929038 | Zbl 1270.90081
[17] J. S. Golan: Semirings and Affine Equations Over Them: Theory and Applications. Mathematics and Its Applications 556. Kluwer Academic, Dordrecht (2003). DOI 10.1007/978-94-017-0383-3 | MR 1997126 | Zbl 1042.16038
[18] M. Gondran, M. Minoux: Graphs, Dioids and Semirings: New Models and Algorithms. Operations Research/Computer Science Interfaces Series 41. Springer, New York (2008). DOI 10.1007/978-0-387-75450-5 | MR 2389137 | Zbl 1201.16038
[19] B. Heidergott, G. J. Olsder, J. van der Woude: Max Plus at Work. Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2006). DOI 10.1515/9781400865239 | MR 2188299 | Zbl 1130.93003
[20] D. Jones: On two-sided max-linear equations. Discrete Appl. Math. 254 (2019), 146-160. DOI 10.1016/j.dam.2018.06.011 | MR 3913099 | Zbl 1407.15017
[21] V. N. Kolokoltsov, V. P. Maslov: Idempotent Analysis and Its Applications. Mathematics and Its Applications 401. Kluwer Academic, Dordrecht (1997). DOI 10.1007/978-94-015-8901-7 | MR 1447629 | Zbl 0941.93001
[22] N. K. Krivulin: Solution of generalized linear vector equations in idempotent algebra. Vestn. St. Petersbg. Univ., Math. 39 (2006), 16-26. MR 2302633
[23] N. Krivulin: A multidimensional tropical optimization problem with a non-linear objective function and linear constraints. Optimization 64 (2015), 1107-1129. DOI 10.1080/02331934.2013.840624 | MR 3316792 | Zbl 1311.65086
[24] N. Krivulin: Extremal properties of tropical eigenvalues and solutions to tropical optimization problems. Linear Algebra Appl. 468 (2015), 211-232. DOI 10.1016/j.laa.2014.06.044 | MR 3293251 | Zbl 1307.65089
[25] N. Krivulin: Algebraic solution of tropical optimization problems via matrix sparsification with application to scheduling. J. Log. Algebr. Methods Program. 89 (2017), 150-170. DOI 10.1016/j.jlamp.2017.03.004 | MR 3634737 | Zbl 1386.90174
[26] N. Krivulin: Complete algebraic solution of multidimensional optimization problems in tropical semifield. J. Log. Algebr. Methods Program. 99 (2018), 26-40. DOI 10.1016/j.jlamp.2018.05.002 | MR 3811166 | Zbl 1412.90143
[27] E. Lorenzo, M. J. de la Puente: An algorithm to describe the solution set of any tropical linear system $A\odot x=B\odot x$. Linear Algebra Appl. 435 (2011), 884-901. DOI 10.1016/j.laa.2011.02.014 | MR 2807241 | Zbl 1217.65076
[28] S. Sergeev: Multiorder, Kleene stars and cyclic projectors in the geometry of max cones. Tropical and Idempotent Mathematics. Contemporary Mathematics 495. American Mathematical Society, Providence, 2009, 317-342. DOI 10.1090/conm/495 | MR 2581526 | Zbl 1179.15033
[29] S. Sergeev, E. Wagneur: Basic solutions of systems with two max-linear inequalities. Linear Algebra Appl. 435 (2011), 1758-1768. DOI 10.1016/j.laa.2011.02.033 | MR 2810669 | Zbl 1227.15018
[30] E. Wagneur, L. Truffet, F. Faye, M. Thiam: Tropical cones defined by max-linear inequalities. Tropical and Idempotent Mathematics. Contemporary Mathematics 495. American Mathematical Society, Providence (2009), 351-366. DOI 10.1090/conm/495 | MR 2581528 | Zbl 1357.15017
[31] M. Yoeli: A note on a generalization of Boolean matrix theory. Am. Math. Mon. 68 (1961), 552-557. DOI 10.2307/2311149 | MR 0126472 | Zbl 0115.02103
[32] K. Zimmermann: A general separation theorem in extremal algebras. Ekon.-Mat. Obz. 13 (1977), 179-201. MR 0453607 | Zbl 0365.90127

Affiliations:   Nikolai Krivulin, St. Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 199034, Russia, e-mail: nkk@math.spbu.ru


 
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