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The Subspace Flatness Conjecture and Faster Integer Programming

Thomas Rothvoss

University of Washington

Abstract: In a seminal paper, Kannan and Lov\’asz (1988) considered a quantity $\mu_{KL}(\Lambda,K)$ which denotes the best volume-based lower bound on the \emph{covering radius} $\mu(\Lambda,K)$ of a convex body $K$ with respect to a lattice $\Lambda$. Kannan and Lov\’asz proved that $\mu(\Lambda,K) \leq n \cdot \mu_{KL}(\Lambda,K)$ and the Subspace Flatness Conjecture by Dadush (2012) claims a $O(\log n)$ factor suffices, which would match the lower bound from the work of Kannan and Lov\’asz.

We settle this conjecture up to a constant in the exponent by proving that  $\mu(\Lambda,K) \leq O(\log^{3}(n)) \cdot \mu_{KL} (\Lambda,K)$. 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 $(\log n)^{O(n)}$-time randomized algorithm to solve integer programs in $n$ variables. Another implication of our main result is a near-optimal \emph{flatness constant} of $O(n \log^{3}(n))$.

Bio: Thomas Rothvoss is Professor of Mathematics & Computer Science at the University of Washington and works on questions in linear and integer programming, discrepancy theory and approximation algorithms. He completed his PhD in Mathematics in 2009 at EPFL under Friedrich Eisenbrand, followed by a PostDoc at MIT with Michel Goemans. He received a STOC 2010 Best Paper Award, a SODA 2014 Best Paper Award, a STOC 2014 Best Paper Award, a FOCS 2023 Best Paper Award, the 2018 Delbert Ray Fulkerson Prize and the 2023 G\”odel Prize. His research is supported by an Alfred P. Sloan Research Fellowship (2015), a David & Lucile Packard Foundation Fellowship (2016) as well as an NSF CAREER Award (2016).