Given a convex function $f$ on $\\mathbb{R}^n$ with an integer minimizer, we show how to find an exact minimizer of $f$ using $O(n^2 \\log n)$ calls to a separation oracle and $O(n^4 \\log n)$ time. The previous best polynomial time algorithm for this problem given in [Jiang, SODA 2021, JACM 2022] achieves $O(n^2\\log\\log n\/\\log n)$ oracle complexity. However, the overall runtime of Jiang’s algorithm is at least $\\widetilde{\\Omega}(n^8)$, due to expensive sub-routines such as the Lenstra-Lenstra-Lov\\’asz (LLL) algorithm [Lenstra, Lenstra, Lov\\’asz, Math. Ann. 1982] and random walk based cutting plane method [Bertsimas, Vempala, JACM 2004]. Our significant speedup is obtained by a nontrivial combination of a faster version of the LLL algorithm due to [Neumaier, Stehl\\’e, ISSAC 2016] that gives similar guarantees, the volumetric center cutting plane method (CPM) by [Vaidya, FOCS 1989] and its fast implementation given in [Jiang, Lee, Song, Wong, STOC 2020].<\/p>\n
For the special case of submodular function minimization (SFM), our result implies a strongly polynomial time algorithm for this problem using $O(n^3 \\log n)$ calls to an evaluation oracle and $O(n^4 \\log n)$ additional arithmetic operations. Both the oracle complexity and the number of arithmetic operations of our more general algorithm are better than the previous best-known runtime algorithms for this specific problem given in [Lee, Sidford, Wong, FOCS 2015] and [Dadush, V\\’egh, Zambelli, SODA 2018, MOR 2021].<\/p>\n","protected":false},"excerpt":{"rendered":"
Given a convex function $f$ on $\\mathbb{R}^n$ with an integer minimizer, we show how to find an exact minimizer of $f$ using $O(n^2 \\log n)$ calls to a separation oracle and $O(n^4 \\log n)$ time. The previous best polynomial time algorithm for this problem given in [Jiang, SODA 2021, JACM 2022] achieves $O(n^2\\log\\log n\/\\log n)$ 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