Conjectured near‑optimal Δ‑parameterized algorithms for ILP-CF and ILP-SF

Establish that, for arbitrary instances of integer linear programming with Δ‑modular constraint matrices A, both (i) the canonical‑form ILP (maximize c^T x subject to A x ≤ b, x ∈ Z^n, where A ∈ Z^{(n+k)×n} has rank n) and (ii) the standard‑form ILP of codimension k (maximize c^T x subject to A x = b, x ∈ Z^n_{≥0}, where A ∈ Z^{k×n} has rank k) admit algorithms whose running time is (log k)^{O(k)} · Δ^2 / 2^{Ω(√log Δ)} + 2^{O(k)} · poly(φ), where Δ is the maximum absolute value of all rank(A)×rank(A) subdeterminants of A and φ denotes the input size; furthermore, determine whether the feasibility variants admit algorithms with running time (log k)^{O(k)} · Δ · (log Δ)^{O(1)} + 2^{O(k)} · poly(φ).

Background

The paper studies two formulations of integer linear programming (ILP) with bounded codimension: ILP in standard form of codimension k (ILP-SF) and ILP in canonical form with n+k constraints (ILP-CF). The authors present improved algorithmic bounds parameterized by the maximum rank-order subdeterminant Δ and by k, including optimization and feasibility variants, via discrepancy arguments and fast tropical/boolean convolution on Abelian groups.

They prove two main theorems (Theorems \ref{main_ILP_th} and \ref{main_ILP_logD_th}) yielding algorithms with running times depending on f_{k,d} or g_{k,Δ}, and show that when n = 2^{(log k)^{O(1)}} or Δ = 2^{(log k)^{O(1)}}, the bounds simplify to forms with only (log k)^{O(k)} dependence. Motivated by these partial results, they explicitly propose the conjecture that such simplified bounds should hold in general, for all n and Δ, for both optimization and feasibility variants of ILP-CF and ILP-SF.

The conjecture targets near-optimal dependence on Δ, consistent with conditional lower bounds via tropical convolution and SETH-based barriers discussed in the paper, and would unify the improved complexities across both ILP formulations without additional dimensional factors.