Better and simpler reducibility bounds over the integers

We study the settings where we are given a function of n variables defined in a given box of integers. We show that in many cases we can replace the given objective function by a new function with a much smaller domain (though still exponential in n). Our approach allows us to transform a family of weakly polynomial upper bounds on analysis measures of algorithms into strongly polynomial ones. To prove our results we use standard tools of linear programming (LP) of a simple LP formulation of the problem of finding an equivalent function of minimum domain size. Our method allows us to derive specific improved bounds for classes of functions and we demonstrate it here on four natural classes of functions. Our bounds are constructive and can be applied both for cases where the goal is to define the equivalent function explicitly or if the goal is to obtain a bound on the domain of a function with equal comparison oracle (as the original function). We note that for the cases considered also in earlier works, we provide here improved upper bounds on the domain of the resulting equivalent function.