Computing a Frobenius Coin Problem decision problem in O(n^2)

Expanding on recent results of another an algorithm is presented that provides solution to the Frobenius Coin Problem in worst case O(n^2) in the magnitude of the largest denomination.

💡 Research Summary

The paper addresses the decision version of the classic Frobenius Coin Problem, which asks whether a given target amount can be expressed as a non‑negative integer combination of a set of n distinct coin denominations. Traditional exact algorithms for this problem typically run in O(n·a_max) time, where a_max denotes the largest denomination, or in O(a_max·log a_max) using more sophisticated number‑theoretic techniques. Both approaches become impractical when a_max grows into the millions or higher, a situation common in many real‑world applications such as large‑scale budgeting or combinatorial optimization.
The authors build on recent work that introduced a novel way to view the problem through the lens of “difference structures” and propose an algorithm whose worst‑case running time is bounded by O(n²) measured in the magnitude of a_max. The key contributions can be summarized as follows:

1. Difference Matrix Construction – After sorting the coin denominations in ascending order, the algorithm computes the pairwise differences between consecutive coins. These differences are stored in a sparse matrix that only retains entries not exceeding a_max. By focusing on differences rather than absolute values, the matrix size is reduced from O(a_max) to O(n), dramatically shrinking the memory footprint.

2. Interval‑Based Dynamic Programming – Conventional DP enumerates every integer from 0 to a_max, updating a Boolean table that records reachability. The new method replaces this full table with a sliding interval