Book

P.Butkovic: Max-linear Systems: Theory and Algorithms (Springer Monographs in Mathematics, Springer-Verlag 2010) Springerlink

Peer-reviewed papers

P.Butkovic and D.Jones: On special cases of the generalized max-plus eigenproblem, SIAM Journal on Matrix Analysis and Applications, Vol.37, No.3 (2016) 1002-1021. DOI: 10.1137/15M1041031.

P.Butkovic: On tropical supereigenvectors, Linear Algebra and Its Applications 498 (2016) 574-591.  DOI: 10.1016/j.laa.2016.02.033.

P.Butkovic and M.MacCaig: A strongly polynomial method for solving integer max-linear programs in a generic case. Journal of Optimization Theory and Applications (2015), Volume 165, Issue 3, pp 941-963. DOI: 10.1007/s10957-014-0596-5 

P.Butkovic: Weakly and strongly stable max-plus matrices, 21st International Symposium on Mathematical Theory of Networks and Systems, July 7-11, 2014. Groningen, The Netherlands.  Extended abstract.

P.Butkovic, H.Schneider and S.Sergeev: On the max-algebraic core of a nonnegative matrix. Electronic Journal of Linear Algebra 27 (2014).

P.Butkovic and M.MacCaig: On the integer max-linear programming problem, Discrete Applied Mathematics 162 (2014) 128–141, DOI: 10.1016/j.dam.2013.08.007

P.Butkovic, H.Schneider, S.Sergeev and B.-S. Tam: Two cores of a nonnegative matrix, Linear Algebra and its Applications 439 (2013) 1929–1954, DOI:10.1016/j.laa.2013.05.029

P.Butkovic and M.MacCaig: On integer eigenvectors and subeigenvectors in the max-plus algebra, Linear Algebra and its Applications 438 (2013) 3408–3424.

P.Butkovic, H.Schneider and S.Sergeev: Recognizing weakly stable matrices, SIAM J. Control Optim. 2012, 50(5), 3029-3051.

P.Butkovic, H.Schneider and S.Sergeev: Z-matrix equations in max algebra, nonnegative linear algebra and other semirings, Linear and Multilinear Algebra 2012, pp 1-20, DOI:10.1080/03081087.2012.656107.

A.Aminu and P.Butkovič: Non-linear programs with max-linear constraints: A heuristic approach, IMA Journal of Management Mathematics 2011; doi: 10.1093/imaman/dpq020.

S.Sergeev, H.Schneider and P.Butkovic: On visualisation scaling, subeigenvectors and Kleene stars in max algebra, Linear Algebra and its Applications 431 (2009) 2395–2406.

P.Butkovic, R.A.Cuninghame-Green and S.Gaubert: Reducible spectral theory with applications to the robustness of matrices in max-algebra, SIAM Journal on Matrix Analysis and Applications 31(3) (2009) 1412-1431.

P.Butkovič and K.P.Tam: On some properties of the image set of a max-linear mapping, Contemporary Mathematics Series, AMS Providence, 495 (2009) 115-126.

P.Butkovič and A.Aminu: Max-linear programming, IMA Journal of Management Mathematics (2009) 20 (3): 233-249.

R.A.Cuninghame-Green and P.Butkovic: Generalised eigenproblem in max algebra, IEEE Xplore, Discrete Event Systems (2008), 9th International Workshop WODES 2008, 236-241, http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=4605951&isnumber=4605912.

P.Butkovic: Finding a bounded mixed-integer solution to a system of dual inequalities, Operations Research Letters 36 (2008) 623-627.

P.Butkovic: Permuted max-algebraic (tropical) eigenvector problem is NP-complete. Linear Algebra and its Applications 428 (2008) 1874-1882.

P.Butkovic: A note on the parity assignment problem. Author Posting. © Taylor & Francis, 2007. This is the author's version of the work. It is posted here by permission of Taylor & Francis for personal use, not for redistribution. The definitive version was published in Optimization, Volume 56, Issue 4, August 2007, pages 419 - 424.

P.Butkovic and P.Szabo: An algorithm for checking strong regularity of matrices in bottleneck algebras. Proceedings of the International Conference Mathematical Methods in Economics and Industry (Herlany) 2007.

P.Butkovic and S.Lewis: On the job rotation problem. Discrete Optimization 4 (2007) 163-174.

P.Butkovic and R.A.Cuninghame-Green: On matrix powers in max-algebra. Linear Algebra and Its Applications 421 (2007) 370-381.

P.Butkovic, H.Schneider and S.Sergeev: Generators, Extremals and Bases of Max Cones. Linear Algebra and Its Applications 421 (2007) 394-406.

P.Butkovic and H.Schneider: Applications of max-algebra to diagonal scaling of matrices. Electronic Journal of Linear Algebra 13 (2005) 262-273.

P.Butkovic: On the combinatorial aspects of max-algebra. In: Idempotent Mathematics and Mathematical Physics (G.L.Litvinov and V.P.Maslov, eds.), Contemporary Mathematics Series, Vol.377, ISSN: 0271-4132 (2005).

R.A.Cuninghame-Green and P.Butkovic: Bases in max-algebra. Linear Algebra and Its Applications 389 (2004) 107-120.

P.Butkovic and R.A.Cuninghame-Green: The linear assignment problem for special matrices. IMA journal of Management Mathematics (2004) 15, 1-12.

R.E.Burkard and P.Butkovic: Max algebra and the linear assignment problem, Mathematical Programming, Ser.B 98: 415-429 (2003).

R.E.Burkard and P.Butkovic: Finding all essential terms of a characteristic maxpolynomial, Discrete Applied Mathematics 130 (2003) 367-380. ISSN: 0166-218X.

P.Butkovic: Max-algebra: the linear algebra of combinatorics? Linear Algebra and Its Applications 367 (2003) 313-335.

P.Butkovic: On the coefficients of max-algebraic characteristic polynomial and characteristic equation. Kybernetika 39 (2003) 129-136, No. 2.

R.A.Cuninghame-Green and P. Butkovic: The equation Ax=By over (max, +), Theoretical Computer Science 293 (2003) 3-12.

P.Butkovic: Simple image set of (max, +) linear mappings, Discrete Applied Mathematics 105 (2000) 73-86. ISSN: 0166-218X.

P.Butkovic and L.Murfitt: Calculating the essential terms of the characteristic maxpolynomials, Central European Journal of Operations Research 8 (2000) 237-246.

P.Butkovic and S.Gaubert: Sign-nonsingular matrices and matrices with unbalanced determinant in symmetrised semirings, Linear Algebra and Its Applications 301 (1999) 195-201.

S.Gaubert, P.Butkovic and R.A.Cuninghame-Green: Minimal (max, +) realization of convex sequences, SIAM J. Control and Optimization 36, No. 1 (1998) 137-147.

R.A.Cuninghame-Green and P.Butkovic: Discrete-event dynamic systems: the strictly convex case, Annals of Oper. Res. 57 (1995) 45-63.

P.Butkovic: Regularity of matrices in min-algebra and its time-complexity, Discrete Applied Mathematics 57 (1995) 121-132.

P.Butkovic: Strong regularity of matrices – a survey of results, Discrete Applied Mathematics 48 (1994) 45-68.

R.A.Cuninghame-Green and P.Butkovic: Extremal eigenproblem for bivalent matrices, Linear Algebra and Its Applications 222 (1995) 77-89.

P.Butkovic and R.A.Cuninghame-Green: An O(n^2) algorithm for the maximum cycle mean of an n×n bivalent matrix, Discrete Appl. Math. 35 (1992) 157-162.

P.Butkovic and R.A.Cuninghame-Green: On the regularity of matrices in min-algebra, Linear Algebra and Its Applications 145 (1991) 127-139.

P.Butkovic and J.Plavka: On the dependence of the maximum cycle mean of a matrix on permutations of the rows and columns, Discrete Applied Mathematics 23 (1989) 45-53.

P.Butkovic, K.Cechlárová and P.Szabó: Strong linear independence in bottleneck algebra, Linear Algebra and Its Applications 94 (1987) 133-155.

P.Butkovic and F.Hevery: A condition for the strong regularity of matrices in the minimax algebra, Discrete Applied Mathematics 11 (1985) 209-222.

P.Butkovic: Necessary solvability conditions of systems of linear extremal equations, Discrete Applied Mathematics 10 (1985) 19-26.

P.Butkovic and G.Hegedüs: An elimination method for finding all solutions of the system of linear equations over an extremal algebra, Ekon. – Mat. Obzor 20 (1984) 203-214.

P.Butkovic: On properties of solution sets of extremal linear programs, Ann. Discrete Math. 19 (1984) 41-54.

P.Butkovic: Solution of systems of linear extremal equations, Ekon. – Mat. Obzor 17 (1981) 402-416.

P.Butkovic: On certain properties of the systems of linear extremal equations, Ekon. – Mat. Obzor 14 (1978) 72-78.

Preprints (unpublished)

P.Butkovic, M.MacCaig:  On integer and extended integer solutions to max-linear systems, School of Mathematics, University of Birmingham, preprint 2015/1.

P.Butkovic, Supereigenvectors, School of Mathematics, University of Birmingham, preprint 2012/3.

P.Butkovic, M.MacCaig: The alternating method for finding integer solutions to two-sided systems, School of Mathematics, University of Birmingham, preprint 2012/8.

P.Butkovic: Permuted linear system problem and permuted eigenvector problem are NP-complete, University of Birmingham, preprint 2008/28.

P.Butkovic, Narrowing the search for eigenvalues in the generalised eigenproblem in max-algebra, School of Mathematics, University of Birmingham, preprint 2008/25.

A.Aminu and P.Butkovič: Comparison of methods for solving two-sided systems in max-algebra, School of Mathematics, University of Birmingham, preprint 2008/08.

P.Butkovic and H.Schneider: On the visualisation scaling of matrices, School of Mathematics, University of Birmingham, preprint 2007/12.

P.Butkovic and M.Fiedler: Tropical tensor product, School of Mathematics, University of Birmingham, preprint 2011/02 .

Other publications

P.Butkovic, Introduction to max-algebra, American Institute of Mathematics, Palo Alto, 2008.

P.Butkovic, L. Hogben, R. Nabben, Z. Strakos and M. Tuma: A brief biography and appreciation of Miroslav Fiedler with a bibliography of his books and papers, Linear Algebra and Its Applications 421 (2007) 173-181.

P.Butkovic: Max-algebra and combinatorial optimisation: connections and open problems. Research Report, Conference Combinatorial Optimization, Oberwolfach 2005.

P.Butkovic: Interactive Excel Investigations in Management Mathematics, MSOR Connections, Vol.2, No.3, August 2002.

P.Butkovic: On the complexity of computing the coefficients of max-algebraic characteristic polynomial and characteristic equation. Proceedings of IFAC 2001 (Prague).

P.Butkovic: Regularity of Matrices, Assignment Problems, Time-Complexity, 26ème école de printemps d’informatique théorique, Noirmoutier 1998 (259-280).