Journal of Global Optimization

, Volume 56, Issue 4, pp 1425–1440 | Cite as

On the determinant and its derivatives of the rank-one corrected generator of a Markov chain on a graph

  • J. A. Filar
  • M. HaythorpeEmail author
  • W. Murray


We present an algorithm to find the determinant and its first and second derivatives of a rank-one corrected generator matrix of a doubly stochastic Markov chain. The motivation arises from the fact that the global minimiser of this determinant solves the Hamiltonian cycle problem. It is essential for algorithms that find global minimisers to evaluate both first and second derivatives at every iteration. Potentially the computation of these derivatives could require an overwhelming amount of work since for the Hessian N 2 cofactors are required. We show how the doubly stochastic structure and the properties of the objective may be exploited to calculate all cofactors from a single LU decomposition.


Markov chain Generator matrix Derivative Determinant Doubly stochastic LU decomposition Rank-one correction Hamiltonian cycle problem Cofactors 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ejov V., Filar J.A., Murray W., Nguyen G.T.: Determinants and longest cycles of graphs. SIAM J. Discret. Math. 22(3), 1215–1225 (2009)CrossRefGoogle Scholar
  2. 2.
    Filar J.A., Krass D.: Hamiltonian cycles and Markov chains. Math. Oper. Res. 19, 223–237 (1994)CrossRefGoogle Scholar
  3. 3.
    Garey M.R., Johnson D.S., Tarjan R.E.: The planar Hamiltonian circuit problem is NP-complete. SIAM J. Comput. 5(4), 704–714 (1976)CrossRefGoogle Scholar
  4. 4.
    Haythorpe, M.: Markov Chain Based Algorithms for the Hamiltonian Cycle Problem. PhD thesis, University of South Australia, 2010. Available at:
  5. 5.
    Heyman D.: A decomposition theorem for infinite stochastic matrices. J. Appl. Probab. 32, 893–901 (1995)CrossRefGoogle Scholar
  6. 6.
    May K.: Derivatives of determinants and other multilinear functions. Math. Mag. 38(5), 207–208 (1965)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2012

Authors and Affiliations

  1. 1.Flinders UniversityAdelaideAustralia
  2. 2.Flinders UniversityAdelaideAustralia
  3. 3.Stanford UniversityStanfordUSA

Personalised recommendations