The Subspace Flatness Conjecture and Faster Integer Programming
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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) ≤ n ·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) ≤ O(log^3(2n)) ·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)).
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