Graph problems arising from parameter identification of discrete dynamical systems

  • Steffen Borchers
  • Sandro BosioEmail author
  • Rolf Findeisen
  • Utz-Uwe Haus
  • Philipp Rumschinski
  • Robert Weismantel
Original Article


This paper focuses on combinatorial feasibility and optimization problems that arise in the context of parameter identification of discrete dynamical systems. Given a candidate parametric model for a physical system and a set of experimental observations, the objective of parameter identification is to provide estimates of the parameter values for which the model can reproduce the experiments. To this end, we define a finite graph corresponding to the model, to each arc of which a set of parameters is associated. Paths in this graph are regarded as feasible only if the sets of parameters corresponding to the arcs of the path have nonempty intersection. We study feasibility and optimization problems on such feasible paths, focusing on computational complexity. We show that, under certain restrictions on the sets of parameters, some of the problems become tractable, whereas others are NP-hard. In a similar vein, we define and study some graph problems for experimental design, whose goal is to support the scientist in optimally designing new experiments.


Graph problems Computational complexity Dynamical systems Parameter identification 


  1. Atkinson AC, Donev AN (1992) Optimum experimental designs. Oxford statistical science series. Oxford University Press, OxfordGoogle Scholar
  2. Bauer I, Bock HG, Körkel S, Schlöder JP (2000) Numerical methods for optimum experimental design in DAE systems. J Comput Appl Math 120(1–2): 1–25MathSciNetzbMATHCrossRefGoogle Scholar
  3. Borchers S, Rumschinski P, Bosio S, Weismantel R, Findeisen R (2009) Model discrimination and parameter estimation for dynamical biochemical reaction networks. In: 15th IFAC symposium on system identificationGoogle Scholar
  4. Chazelle B (1980) Computational geometry and convexity. Ph.D. thesis, Yale UniversityGoogle Scholar
  5. Chazelle B (1984) Convex partitions of polyhedra: a lower bound and worst-case optimal algorithm. SIAM J Comput 13(3): 488–507MathSciNetzbMATHCrossRefGoogle Scholar
  6. Chazelle B (1986) Filtering search: a new approach to query-answering. SIAM J Comput 15(3): 703–724MathSciNetzbMATHCrossRefGoogle Scholar
  7. Chazelle B (1988) A functional approach to data structures and its use in multidimensional searching. SIAM J Comput 17(3): 427–462MathSciNetzbMATHCrossRefGoogle Scholar
  8. Chazelle B (1993) An optimal convex hull algorithm in any fixed dimension. Discret Comput Geom 10: 377–409MathSciNetzbMATHCrossRefGoogle Scholar
  9. Chazelle B, Dobkin DP. (1985) Optimal convex decompositions. In: Toussaint GT (eds) Computational Geometry. North-Holland, Amsterdam, pp 63–133Google Scholar
  10. Culberson JC, Reckhow RA (1989) Orthogonally convex coverings of orthogonal polygons without holes. J Comput Syst Sci 39(2): 166–204MathSciNetzbMATHCrossRefGoogle Scholar
  11. Edelsbrunner H (1983) A new approach to rectangle intersections. I, II Int J Comput Math 13(3–4):209–219, 221–229Google Scholar
  12. Edelsbrunner H, Haring G, Hilbert D (1986) Rectangular point location in d dimensions with applications. Comput J 29(1): 76–82MathSciNetCrossRefGoogle Scholar
  13. Edelsbrunner H, O’Rourke J, Seidel R (1986) Constructing arrangements of lines and hyperplanes with applications. SIAM J Comput 15(2): 341–363MathSciNetzbMATHCrossRefGoogle Scholar
  14. Evans ND, Chappell MJ, Chapman MJ, Godfrey KR (2004) Structural indistinguishability between uncontrolled (autonomous) nonlinear analytic systems. Automatica 40: 1947–1953MathSciNetzbMATHCrossRefGoogle Scholar
  15. Håstad J (1999) Clique is hard to approximate within n 1-ε. Acta Math 182(1): 105–142MathSciNetzbMATHCrossRefGoogle Scholar
  16. Kuepfer L, Sauer U, Parrilo PA (2007) Efficient classification of complete parameter regions based on semidefinite programming. BMC Bioinf 8: 12CrossRefGoogle Scholar
  17. Lingas A (1982) The power of non-rectilinear holes. In: 9th colloquium on automata, languages and programming, Lecture notes in computer science, vol 140. pp 369–383Google Scholar
  18. Marquardt DW (1963) An algorithm for least-squares of nonlinear parameters. SIAM J Appl Math 11: 431–441MathSciNetzbMATHCrossRefGoogle Scholar
  19. McMullen P (1970) The maximum numbers of faces of a convex polytope. Mathematika 17: 179–184MathSciNetzbMATHCrossRefGoogle Scholar
  20. O’Rourke J, Supowit KJ (1983) Some NP-hard polygon decomposition problems. IEEE Trans Inf Theory 29(2): 181–190MathSciNetzbMATHCrossRefGoogle Scholar
  21. Pázman A (1986) Foundations of optimum experimental design. Reidel, DordrechtzbMATHGoogle Scholar
  22. Prajna S (2006) Barrier certificates for nonlinear model validation. Automatica 42: 117–126MathSciNetzbMATHCrossRefGoogle Scholar
  23. Preparata F, Shamos MI (1985) Computational geometry. Spinger, New YorkGoogle Scholar
  24. Ramsay JO, Hooker G, Campbell D, Cao J (2007) Parameter estimation for differential equations: a generalized smoothing approach. J R Stat Soc Ser B 69(5): 741–796MathSciNetCrossRefGoogle Scholar
  25. Robert CP, Casella G (2004) Monte Carlo statistical methods. Springer, BerlinzbMATHGoogle Scholar
  26. Schittkowski K (2007) Experimental design tools for ordinary and algebraic differential equations. Ind Eng Chem Res 46(26): 9137–9147CrossRefGoogle Scholar
  27. Schnell S, Chappell MJ, Evans ND, Roussel MR (2006) The mechanism distinguishability problem in biochemical kinetics: the single-enzyme, single-substrate reaction as a case study. CR Biol 329: 51–61CrossRefGoogle Scholar
  28. Schrijver A (2003) Combinatorial optimization. Polyhedra and efficiency. Algorithms and combinatorics, vol 24. Springer-Verlag, BerlinGoogle Scholar
  29. Six HW, Wood D (1982) Counting and reporting intersections of d-ranges. IEEE Trans Comput 31(3): 181–187MathSciNetzbMATHCrossRefGoogle Scholar
  30. Smith RS, Doyle JC (1992) Model validation: a connection between robust control and identification. IEEE Trans Autom Control 37(7): 942–952MathSciNetzbMATHCrossRefGoogle Scholar
  31. Tomlin JA, Welch JS (1986) Finding duplicate rows in a linear programming model. Oper Res Lett 5(1): 7–11MathSciNetzbMATHCrossRefGoogle Scholar
  32. Valiant LG, Vazirani VV (1985) NP is as easy as detecting unique solutions. In: 17th ACM symposium on theory of computing. pp 458–463Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Steffen Borchers
    • 1
  • Sandro Bosio
    • 2
    Email author
  • Rolf Findeisen
    • 1
  • Utz-Uwe Haus
    • 2
  • Philipp Rumschinski
    • 1
  • Robert Weismantel
    • 2
  1. 1.Institut für AutomatisierungstechnikOtto-von-Guericke UniversitätMagdeburgGermany
  2. 2.Institute for Operations Research (IFOR)Eidgenössischen Technischen Hochschule (ETH) ZürichZürichSwitzerland

Personalised recommendations