Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Faster Algorithms for Frobenius Numbers. | Faster Algorithms for Frobenius Numbers Dale Beihoffer Lakeville Minnesota USA dbeihoffer@ Jemimah Hendry Madison Wisconsin USA jhendry@ Albert Nijenhuis Seattle Washington USA nijenhuis@ Stan Wagon Macalester College St. Paul Minnesota USA wagon@ Submitted Oct 10 2004 Accepted May 3 2005 Published Jun 14 2005 Mathematics Subject Classifications 05C85 11Y50 Abstract The Frobenius problem also known as the postage-stamp problem or the moneychanging problem is an integer programming problem that seeks nonnegative integer solutions to x1a1 xnan M where ai and M are positive integers. In particular the Frobenius number f A where A aig 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 a1 107. 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 a1 pn . The previous best method is due to Bocker and Lipták and runs in time O a1n . 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 1 n nA 1 ra 1 EA. THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R27 1 1. .
