Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Atkinson AC, Donev AN (1992) Optimum experimental designs. Oxford statistical science series. Oxford University Press, Oxford
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): 1–25
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 identification
Chazelle B (1980) Computational geometry and convexity. Ph.D. thesis, Yale University
Chazelle B (1984) Convex partitions of polyhedra: a lower bound and worst-case optimal algorithm. SIAM J Comput 13(3): 488–507
Chazelle B (1986) Filtering search: a new approach to query-answering. SIAM J Comput 15(3): 703–724
Chazelle B (1988) A functional approach to data structures and its use in multidimensional searching. SIAM J Comput 17(3): 427–462
Chazelle B (1993) An optimal convex hull algorithm in any fixed dimension. Discret Comput Geom 10: 377–409
Chazelle B, Dobkin DP. (1985) Optimal convex decompositions. In: Toussaint GT (eds) Computational Geometry. North-Holland, Amsterdam, pp 63–133
Culberson JC, Reckhow RA (1989) Orthogonally convex coverings of orthogonal polygons without holes. J Comput Syst Sci 39(2): 166–204
Edelsbrunner H (1983) A new approach to rectangle intersections. I, II Int J Comput Math 13(3–4):209–219, 221–229
Edelsbrunner H, Haring G, Hilbert D (1986) Rectangular point location in d dimensions with applications. Comput J 29(1): 76–82
Edelsbrunner H, O’Rourke J, Seidel R (1986) Constructing arrangements of lines and hyperplanes with applications. SIAM J Comput 15(2): 341–363
Evans ND, Chappell MJ, Chapman MJ, Godfrey KR (2004) Structural indistinguishability between uncontrolled (autonomous) nonlinear analytic systems. Automatica 40: 1947–1953
Håstad J (1999) Clique is hard to approximate within n 1-ε. Acta Math 182(1): 105–142
Kuepfer L, Sauer U, Parrilo PA (2007) Efficient classification of complete parameter regions based on semidefinite programming. BMC Bioinf 8: 12
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 369–383
Marquardt DW (1963) An algorithm for least-squares of nonlinear parameters. SIAM J Appl Math 11: 431–441
McMullen P (1970) The maximum numbers of faces of a convex polytope. Mathematika 17: 179–184
O’Rourke J, Supowit KJ (1983) Some NP-hard polygon decomposition problems. IEEE Trans Inf Theory 29(2): 181–190
Pázman A (1986) Foundations of optimum experimental design. Reidel, Dordrecht
Prajna S (2006) Barrier certificates for nonlinear model validation. Automatica 42: 117–126
Preparata F, Shamos MI (1985) Computational geometry. Spinger, New York
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): 741–796
Robert CP, Casella G (2004) Monte Carlo statistical methods. Springer, Berlin
Schittkowski K (2007) Experimental design tools for ordinary and algebraic differential equations. Ind Eng Chem Res 46(26): 9137–9147
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: 51–61
Schrijver A (2003) Combinatorial optimization. Polyhedra and efficiency. Algorithms and combinatorics, vol 24. Springer-Verlag, Berlin
Six HW, Wood D (1982) Counting and reporting intersections of d-ranges. IEEE Trans Comput 31(3): 181–187
Smith RS, Doyle JC (1992) Model validation: a connection between robust control and identification. IEEE Trans Autom Control 37(7): 942–952
Tomlin JA, Welch JS (1986) Finding duplicate rows in a linear programming model. Oper Res Lett 5(1): 7–11
Valiant LG, Vazirani VV (1985) NP is as easy as detecting unique solutions. In: 17th ACM symposium on theory of computing. pp 458–463
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Borchers, S., Bosio, S., Findeisen, R. et al. Graph problems arising from parameter identification of discrete dynamical systems. Math Meth Oper Res 73, 381–400 (2011). https://doi.org/10.1007/s00186-011-0356-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00186-011-0356-3