On $$\Delta $$-modular integer linear problems in the canonical form and equivalent problems

Many papers in the field of integer linear programming (ILP, for short) are devoted to problems type $$\max \{c^\top x :A = b,\, \in {{\,\mathrm{\mathbb {Z}}\,}}^n_{\ge 0}\}$$ , where all entries A, b, c integer, parameterized by number rows A and $$\Vert A\Vert _{\max }$$ . This class is known under name ILP standard form, adding word ”bounded” if $$x \le u$$ some vector u. Recently, many new sparsity, proximity, complexity results were obtained bounded unbounded form. In this paper, we consider canonical form $$\begin{aligned} \max :b_l b_r,\, {Z}}\,}}^n\}, \end{aligned}$$ $$b_l$$ $$b_r$$ vectors. We assume that matrix has rank n, $$(n + m)$$ rows, n columns, parameterize problem m $$\Delta (A)$$ maximum $$n \times n$$ sub-determinants taken absolute value. show any can be polynomially reduced preserving but reverse reduction not always possible. More precisely, define generalized which includes an additional group constraint, prove equivalence generalize bounds Additionally, sometimes, strengthen previously and, give shorter proofs. Finally, special cases $$m \{0,1\}$$ By way, specialised on simplices, Knapsack Subset-Sum problems.