The Subspace Flatness Conjecture and Faster Integer Programming

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 $O(n \log^{2}(2n))$, improving on the previous bound of $O(n^{4/3} \log^{O(1)} (2n))$.