{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:44:57Z","timestamp":1760237097674,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,2,21]],"date-time":"2020-02-21T00:00:00Z","timestamp":1582243200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we consider the eigenproblems for Latin squares in a bipartite min-max-plus system. The focus is upon developing a new algorithm to compute the eigenvalue and eigenvectors (trivial and non-trivial) for Latin squares in a bipartite min-max-plus system. We illustrate the algorithm using some examples. The proposed algorithm is implemented in MATLAB, using max-plus algebra toolbox. Computationally speaking, our algorithm has a clear advantage over the power algorithm presented by Subiono and van der Woude. Because our algorithm takes     0 . 088783     sec to solve the eigenvalue problem for Latin square presented in Example 2, while the compared one takes     1 . 718662     sec for the same problem. Furthermore, a time complexity comparison is presented, which reveals that the proposed algorithm is less time consuming when compared with some of the existing algorithms.<\/jats:p>","DOI":"10.3390\/sym12020311","type":"journal-article","created":{"date-parts":[[2020,2,26]],"date-time":"2020-02-26T04:18:29Z","timestamp":1582690709000},"page":"311","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["An Efficient Algorithm for Eigenvalue Problem of Latin Squares in a Bipartite Min-Max-Plus System"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6523-4316","authenticated-orcid":false,"given":"Mubasher","family":"Umer","sequence":"first","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Umar","family":"Hayat","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fazal","family":"Abbas","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Sciences, Stetson University, DeLand, FL 32723, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Anurag","family":"Agarwal","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY 14623, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Petko","family":"Kitanov","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Wells College, Aurora, NY 13026, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1185","DOI":"10.1007\/s10587-006-0087-4","article-title":"Invertible commutativity preservers of matrices over max algebra","volume":"56","author":"Song","year":"2006","journal-title":"Czechoslovak Math. J."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1649","DOI":"10.1080\/03081087.2014.964168","article-title":"Regular matrices and their generalized inverses over the max algebra","volume":"63","author":"Kang","year":"2015","journal-title":"Linear Multilinear Alg."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1080\/0308108031000069173","article-title":"Column ranks and their preservers of matrices over max algebra","volume":"51","author":"Song","year":"2003","journal-title":"Linear Multilinear Alg."},{"key":"ref_4","first-page":"887","article-title":"Tropical nevanlinna theory and ultra-discrete equations","volume":"5","author":"Halburd","year":"2009","journal-title":"Int. Math. Res. Not."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Cuninghame-Green, R.A. (1979). Lecture notes in economics and mathematical systems. 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World J., 6718653.","DOI":"10.1155\/2018\/6718653"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"278","DOI":"10.1016\/j.dam.2018.03.045","article-title":"Applications of max-plus algebra to flow shop scheduling problems","volume":"247","author":"Kubo","year":"2018","journal-title":"Discret. Appl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/0024-3795(93)90492-7","article-title":"The power algorithm in max algebra","volume":"182","author":"Braker","year":"1993","journal-title":"Linear Alg. Appl."},{"key":"ref_11","unstructured":"(2000). On Classes of Min-Max-Plus Systems and Their Application. [Ph.D. Thesis, Delft University of Technology]."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1023\/A:1008315821604","article-title":"Power algorithms for (max,+)- and bipartite (min,max,+)-systems","volume":"10","author":"Subiono","year":"2000","journal-title":"Discret. Event Dyn. Syst."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Umer, M., Hayat, U., and Abbas, F. (2019). An efficient algorithm for nontrivial eigenvectors in max-plus algebra. Symmetry, 11.","DOI":"10.3390\/sym11060738"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1007\/s10626-014-0204-8","article-title":"Eigenproblems of latin squares in bipartite (min, max, plus)-systems","volume":"26","author":"Subiono","year":"2016","journal-title":"Discret. Event Dyn. Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3261","DOI":"10.1016\/j.laa.2011.06.009","article-title":"Best approximation in maxplus semimodules","volume":"435","author":"Akian","year":"2011","journal-title":"Linear Alg. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"390","DOI":"10.4236\/apm.2015.57038","article-title":"Eigenvectors of permutation matrices","volume":"5","author":"Magret","year":"2015","journal-title":"Adv. Pure Math."},{"key":"ref_17","unstructured":"Sta\u0144czyk, J. (2020, February 16). Max-Plus Algebra Toolbox for Matlab. Available online: http:\/\/gen.up.wroc.pl\/stanczyk\/mpa\/."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1007\/s00026-005-0261-7","article-title":"On the number of Latin squares","volume":"9","author":"McKay","year":"2005","journal-title":"Ann. 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