Polynomial Approximation Schemes and Exact Algorithms for Optimum Curve Segmentation Problems
Résumé
Optimum curve segmentation problems typically arise when analyzing data represented by curves or graphs of real-valued functions in one real variable. Examples of applications are many, including: • time series analysis and forecasting; • analysis and identification of dynamical systems; • process-monitoring (analysis and modelling of input-output relationships for physical systems such as chemical reactors, engines, electrical systems, biological systems, etc.).We propose here a general framework for stating and solving such problems, either exactly or approximately, using polynomial approximation schemes. Both the discrete version of the problem (Discrete Segmentation Problem, DSP) and the continuous version of the problem (Continuous Segmentation Problem, CSP) are addressed. We investigate various sets of conditions under which DSP or CSP can be solved either exactly in polynomial time or approximately by means of a fully polynomial-time approximation scheme (FPTAS). Finally, we formulate the discrete segmentation problem with variable number of segments (DSPV) and show that it can be formulated as an integer linear program reducible to minimum cost network flow and shortest path computations.