Polynomial-time solvability of Δ-modular integer programming

Determine whether the integer program max{ c^T x : A^T x ≤ b, x ∈ Z^d } with A ∈ Z^{d×n} having all square subdeterminants bounded by a fixed constant Δ admits a polynomial-time algorithm for every fixed Δ.

Background

The paper reviews the classical role of totally unimodular (TU) matrices, for which integer programs can be solved in polynomial time, and the broader notion of Δ-modular matrices, where all square subdeterminants are bounded by Δ.

While 2-modular integer programs are known to be solvable in polynomial time, and certain additional structural restrictions (e.g., at most two nonzeros per row or per column) also yield polynomial solvability, the general case for any fixed Δ remains unresolved and is highlighted as a prominent open problem.