# On polynomial-time solvable linear Diophantine problems

## TL;DR

This paper presents a polynomial-time algorithm for solving certain linear Diophantine problems with specific matrix structures, improving previous conditions for efficient solvability.

## Contribution

The authors develop a new polynomial-time algorithm for linear Diophantine problems with matrices containing a nonsingular submatrix, expanding the class of problems solvable efficiently.

## Key findings

- Algorithm successfully finds solutions or determines infeasibility in polynomial time.
- Improves previous complexity bounds for specific classes of linear Diophantine problems.
- Applicable when the right-hand side is deep in the cone generated by a submatrix's columns.

## Abstract

We obtain a polynomial-time algorithm that, given input (A, b), where A=(B|N) is an integer mxn matrix, m<n, with nonsingular mxm submatrix B and b is an m-dimensional integer vector, finds a nonnegative integer solution to the system Ax=b or determines that no such solution exists, provided that b is located sufficiently "deep" in the cone generated by the columns of B. This result improves on some of the previously known conditions that guarantee polynomial-time solvability of linear Diophantine problems.