Totally $\Delta$-modular IPs with two non-zeros in most rows

TL;DR

This paper presents a strongly polynomial time algorithm for solving certain integer programs with matrices having bounded subdeterminants and at most two non-zero entries per row after removing some rows and columns, extending previous work.

Contribution

It introduces a new algorithm that handles more general cases of IPs with bounded subdeterminants and limited non-zero entries per row, broadening the scope of polynomial-time solvability.

Findings

Algorithm runs in strongly polynomial time.

Extends previous results to more general IPs.

Handles additional constraints and variables.

Abstract

Integer programs (IPs) on constraint matrices with bounded subdeterminants are conjectured to be solvable in polynomial time. We give a strongly polynomial time algorithm to solve IPs where the constraint matrix has bounded subdeterminants and at most two non-zeros per row after removing a constant number of rows and columns. This result extends the work by Fiorini, Joret, Weltge \& Yuditsky (J. ACM 72(1), 1-50 (2025)) by allowing for additional, unifying constraints and variables.