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Partitioning and grouping of similar objects plays a fundamental role inimage segmentation and in clustering problems. In such problems a typical goalis to group together similar objects, or pixels in the case of imageprocessing. At the same time another goal is to have each group distinctlydissimilar from the rest and possibly to have the group size fairly large.These goals are often combined as a ratio optimization problem. One example ofsuch problem is the normalized cut problem, another is the ratio regionsproblem. We devise here the first polynomial time algorithms solving theseproblems optimally. The algorithms are efficient and combinatorial. Thiscontrasts with the heuristic approaches used in the image segmentationliterature that formulate those problems as nonlinear optimization problems,which are then relaxed and solved with spectral techniques in real numbers.These approaches not only fail to deliver an optimal solution, but they arealso computationally expensive. The algorithms presented here use as asubroutine a minimum $s,t-cut procedure on a related graph which is ofpolynomial size. The output consists of the optimal solution to the respectiveratio problem, as well as a sequence of nested solution with respect to anyrelative weighting of the objectives of the numerator and denominator.  An extension of the results here to bi-criteria and multi-criteria objectivefunctions is presented in part II.

Polynomial time algorithms for bi-criteria, multi-objective and ratio problems in clustering and imaging. Part I: Normalized cut and ratio regions