Fast Fourier Transform and its Applications to Integer Knapsack Problems

In this paper we suggest a new efficient technique for solving integer knapsack problems. Our algorithms can be seen as application of Fast Fourier Transform to generating functions of integer polytopes. Using this approach, it is possible to count the number of boolean solutions of a single n-dimensional Diophantine equation [a,x]= b in O(|a|1 ln|a1|ln n) operations. Another application example is an integer knapsack optimization problem of volume b, which can be solved in O(|a|1 ln|a1|ln n + b ln2 n) operations of exact real arithmetics. These complexity estimates improve by a factor of n the complexity of the traditional Dynamic Programming technique