Properties of $\Delta$-modular simplicies and "close" polyhedrons. $O(\Delta \cdot polylog(\Delta))$-algorithm for knapsack, proximity and sparsity

In this work we consider properties of square and "close"-square $\Delta$-modular systems of linear inequalities $A x \leq b$. More precisely, we study some class $\mathscr{P}$ of "local" polyhedrons defined by $\Delta$-modular systems of the type $A x \leq b$ that includes simplicies, simple cones, parallelotopes (affine images of cubes) and some more general polyhedrons. We show that for $P \in \mathscr{P}$ the Integer Linear Programming (the ILP) problem $\max\{c^\top x \colon x \in P \cap \mathbb{Z}^n\}$ can be solved by an algorithm with the complexity $$O(\Delta \cdot \log \Delta \cdot M + poly(n,s)),$$ where $M = (m-n) \cdot mult(\log \Delta) + mult(\log \|c\|_\infty)$, $s$ is input size and $mult(t)$ is the complexity of $t$-bit integers multiplication. Additionally, we give estimates on proximity and sparsity of a solution, and show that for fixed $A$, with high probability, the system $A x \leq b$ defines polyhedron from $\mathscr{P}$. Another ingredient is a lemma that states equality of rank minors of matricies with orthogonal columns. This lemma gives us an opportunity to transform the systems of the type $A x = b,\, x \geq 0$ to systems of the type $A x \leq b$ and vise verse, such that structure of sub-determinants states the same. By this way, using the mentioned results about properties of the family $\mathscr{P}$, we give an algorithm for the unbounded knapsack problem with the complexity $$O(\Delta \cdot \log \Delta \cdot M + poly(n,s)),$$ where $\Delta = \|a\|_\infty$, $M = mult(\log \Delta) + mult(\log \|c\|_\infty)$. Additionally, we give estimates on the proximity and sparsity of a solution. Finally, using close technics, we show that the number of unimodular equivalence classes of $\Delta$-modular integrally-empty simplicies is bounded by the function $O(\Delta^{3+\log \Delta} \cdot (2n)^\Delta)$. And give an efficient algorithm to enumerate them.