# Properties of Δ-modular simplicies and "close" polyhedrons. O(Δ· polylog(Δ))-algorithm for knapsack, proximity and sparsity

In this work we consider properties of square and "close"-square Δ-modular systems of linear inequalities A x ≤ b. More precisely, we study some class P of "local" polyhedrons defined by Δ-modular systems of the type A x ≤ b that includes simplicies, simple cones, parallelotopes (affine images of cubes) and some more general polyhedrons. We show that for P ∈P the Integer Linear Programming (the ILP) problem max{c^ x x ∈ P ∩Z^n} can be solved by an algorithm with the complexity O(Δ·logΔ· M + poly(n,s)), where M = (m-n) · mult(logΔ) + mult(logc_∞), 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 ≤ b defines polyhedron from 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 ≥ 0 to systems of the type A x ≤ 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 P, we give an algorithm for the unbounded knapsack problem with the complexity O(Δ·logΔ· M + poly(n,s)), where Δ = a_∞, M = mult(logΔ) + mult(logc_∞). 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 Δ-modular integrally-empty simplicies is bounded by the function O(Δ^3+logΔ· (2n)^Δ). And give an efficient algorithm to enumerate them.

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