A fast algorithm for the Frobenius problem in three variables

TL;DR

The paper tackles the Frobenius problem for three variables by building on Brauer–Shockley and Tripathi's theorem to express the Frobenius number g(A) via minima of a carefully crafted function F(n,r). It introduces arithmetic reduced modulo (ARM) sequences and their border/diff-mod structures, enabling efficient computation of the minima across residue classes and guiding a logarithmic-time algorithm in the input size. The authors derive a comprehensive set of closed-form formulas (six mutually exclusive cases) for g(A) that cover all configurations, along with an O(log a1) procedure for computing a key intermediate parameter nb. The approach yields a practical, fast algorithm that matches the best-known three-variable performance and clarifies the role of modular sequences in solving a classic combinatorial number theory problem.

Abstract

Given a set of three positive integers {a1, a2, a3}, denoted A, the Frobenius problem in three variables is to find the greatest integer which cannot be expressed in the following form, where x1, x2 and x3 are non-negative integers: x1*a1 + x2*a2 + x3*a3 The fastest known algorithm for solving the three variable case of the Frobenius problem was invented by H. Greenberg in 1988 whose worst case time complexity is a logarithmic function of A. In 2017 A. Tripathi presented another algorithm for solving the same problem. This article presents an algorithm whose foundation is the same as Tripathi's. However, the algorithm presented here is significantly different from Tripathi's and we show that its worst case time complexity also is a logarithmic function of A