Faster Algorithms for Frobenius Numbers

Abstract

The Frobenius problem, also known as the postage-stamp problem or the money-changing problem, is an integer programming problem that seeks nonnegative integer solutions to $ x_1 a_1 +\cdots + x_n a_n =M$, where $ a_i $ and $M$ are positive integers. In particular, the Frobenius number $f(A)$, where $A=\{ a_i \}$, is the largest $M$ so that this equation fails to have a solution. A simple way to compute this number is to transform the problem to a shortest-path problem in a directed weighted graph; then Dijkstra's algorithm can be used. We show how one can use the additional symmetry properties of the graph in question to design algorithms that are very fast. For example, on a standard desktop computer, our methods can handle cases where $n=10$ and $ a_1 = 10^7$. We have two main new methods, one based on breadth-first search and another that uses the number theory and combinatorial structure inherent in the problem to speed up the Dijkstra approach. For both methods we conjecture that the average-case complexity is $O( a_1 \sqrt{n} )$. The previous best method is due to Böcker and Lipták and runs in time $O( a_1 n)$. These algorithms can also be used to solve auxiliary problems such as (1) find a solution to the main equation for a given value of $M$; or (2) eliminate all redundant entries from a basis. We then show how the graph theory model leads to a new upper bound on $f(A)$ that is significantly lower than existing upper bounds. We also present a conjecture, supported by many computations, that the expected value of $f(A)$ is a small constant multiple of $ \left({1\over2} \, n! \, \Pi A\right)^{1/(n-1)} -\Sigma A$.