The Subspace Flatness Conjecture and Faster Integer Programming
Victor Reis, Thomas Rothvoss
TL;DR
The paper resolves, up to a constant in the exponent, the Subspace Flatness Conjecture by proving μ_KL(Λ,K) ≤ μ(Λ,K) ≤ O(log^3(2n))·μ_KL(Λ,K) for any full-rank lattice Λ and convex body K. The approach builds on the Reverse Minkowski Theorem and a structured filtration of Λ into stable, well-separated pieces, enabling an inductive proof that relates the covering radius to a subspace-projected, volume-based witness. This yields algorithmic consequences: a randomized algorithm to find a witnessing subspace W in time 2^{O(n)} that leads to an IP solver with running time (log(2n))^{O(n)} and a near-optimal flatness constant O(n log^3(2n)). The results also imply that μ(Λ,K−K) provides a near-equivalent bound to μ(Λ,K), improving the known flatness bounds and enhancing the complexity of integer programming in fixed dimension with lattice-oracular access to the convex body.
Abstract
In a seminal paper, Kannan and Lovász (1988) considered a quantity $μ_{KL}(Λ,K)$ which denotes the best volume-based lower bound on the covering radius $μ(Λ,K)$ of a convex body $K$ with respect to a lattice $Λ$. Kannan and Lovász proved that $μ(Λ,K) \leq n \cdot μ_{KL}(Λ,K)$ and the Subspace Flatness Conjecture by Dadush (2012) claims a $O(\log(2n))$ factor suffices, which would match the lower bound from the work of Kannan and Lovász. We settle this conjecture up to a constant in the exponent by proving that $μ(Λ,K) \leq O(\log^{3}(2n)) \cdot μ_{KL} (Λ,K)$. Our proof is based on the Reverse Minkowski Theorem due to Regev and Stephens-Davidowitz (2017). Following the work of Dadush (2012, 2019), we obtain a $(\log(2n))^{O(n)}$-time randomized algorithm to solve integer programs in $n$ variables. Another implication of our main result is a near-optimal flatness constant of $O(n \log^{3}(2n))$.