A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed
Mathematics of Operations Research
Parameterized polyhedra and their vertices
International Journal of Parallel Programming
Parametric Analysis of Polyhedral Iteration Spaces
Journal of VLSI Signal Processing Systems - Special issue on application specific systems, architectures and processors
A Primal Barvinok Algorithm Based on Irrational Decompositions
SIAM Journal on Discrete Mathematics
Presburger arithmetic, rational generating functions, and quasi-polynomials
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part II
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We examine two different ways of encoding a counting function: as a rational generating function and explicitly as a function (defined piecewise using the greatest integer function). We prove that, if the degree and number of input variables of the (quasi-polynomial) function are fixed, there is a polynomial time algorithm which converts between the two representations. Examples of such counting functions include Ehrhart quasi-polynomials, vector partition functions, integer points in parametric polytopes, and projections of the integer points in parametric polytopes. For this last example, this algorithm provides the first known way to compute the explicit function in polynomial time. We rely heavily on results by Barvinok and Pommersheim [Barvinok, A., Pommersheim, J., 1999. An algorithmic theory of lattice points in polyhedra. In: New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-97). In: Math. Sci. Res. Inst. Publ., vol. 38. Cambridge Univ. Press, Cambridge, pp. 91-147], and also by Verdoolaege et al. [Verdoolaege, S., Seghir, R., Beyls, K., Loechner, V., Bruynooghe, M., 2007. Counting integer points in parametric polytopes using Barvinok's rational functions, Algorithmica 48 (1), 37-66].