Integer programming with binary and bounded variables via Gröbner bases with applications to multiobjective integer programming - Mattos Langeloh, Gabriel (Journal of symbolic computation, 2026)
Families of integer programming problems can be solved efficiently in practice once their reduced Gröbner basis is known. However, computing Gröbner bases is often hard, especially when binary or integer bounded variables are present in the problem formulation. In this paper, we study the specific structure of the constraint matrix of integer programs with bounded variables and the implications of this structure to the truncated Gröbner bases of integer programming problems. In this direction, we introduce a new Binary Truncation Criterion that is capable of predicting and eliminating useless S-vectors before they built in the Gröbner basis computation. Additionally, we propose improvements to the Gröbner basis approach to multiobjective integer programming of Jiménez-Tafur (2017) and Hartillo-Hermoso et al. (2020), such as a proof that truncated Gröbner bases can be used in their algorithm with no loss of correctness, implying that our new truncation techniques are also useful in this application. All new proposed methods are implemented in the open source package IPGBs and their performance is empirically validated.