New results for aggregating integer-valued equations

A variety of results have been given for aggregating integer-valued (diophantine) equations whose variables are restricted to nonnegative integers. In each, integer weights are identified for the equations so that their linear combination yields a single equation with the same solution set of the original system of equations. Because the coefficients of the aggregated equation tend to achieve unwieldy sizes as the number of original equations increases, the goal is to identify weights so these coefficients will lie in a range as limited as possible. We give theorems which separately and in combination provide new methods for aggregating general integer-valued equations. Our results include formulations that do not require linearity of the original system, or nonnegativity of component variables. We also demonstrate that our theorems yield as special cases earlier results (analytical formulae) conjectured to yield the smallest possible weights for less general domains. As another application, the presented results were used to develop a highly efficient approach for the integer knapsack problem. Empirical outcomes show that the developed solution procedure is significantly superior to advanced branch and bound methods (previously established to be the most efficient knapsack solution procedures).