Convex Minimization with Integer Minima in $\widetilde O(n^4)$ Time

TL;DR

This work addresses the exact minimization of a convex function $f$ on $\mathbb{R}^n$ when the minimizers are integral, assuming access to a separation oracle. The authors introduce a randomized algorithm that achieves $O(n^2 \log(nR))$ oracle calls and $O(n^4 \log(nR))$ arithmetic operations by combining a block-wise cutting plane method centered on the volumetric center with a fast approximate shortest-vector approach and leverage-score maintenance, plus a dimension-reduction strategy that preserves integrality. For submodular function minimization, the result yields a strongly polynomial-time procedure with $O(n^3 \log n)$ evaluation oracle calls and $O(n^4 \log n)$ arithmetic, improving over prior work in arithmetic efficiency while remaining competitive in oracle complexity. The main methodological innovations are cutting-plane in blocks (lazy width measurement), the use of Vaidya’s volumetric center, and a faster LLL-based SVP routine, enabling substantial runtime speedups over previous strongly-polynomial-time algorithms. The findings have significant practical impact for problems like SFM, where stronger polynomial-time guarantees translate into faster, scalable optimization in discrete settings, especially when integer-optimal solutions are natural or required.

Abstract

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 $\widetildeΩ(n^8)$, due to expensive sub-routines such as the Lenstra-Lenstra-Lovász (LLL) algorithm [Lenstra, Lenstra, Lovász, 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, Stehlé, 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]. 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, Végh, Zambelli, SODA 2018, MOR 2021].

Theorems & Definitions (46)

• Theorem 1.1: Main result, informal version of Theorem \ref{['thm:main_formal']}
• Corollary 1.2: Submodular function minimization
• Definition 3.1: Dual lattice
• Definition 3.2: Leverage score
• Theorem 3.4: LLL algorithm, lll82
• Theorem 3.5: Theorem 2 of ns16
• Definition 3.6
• Definition 3.7: Covariance of convex body, $\mathrm{Cov}(K)$
• Lemma 3.8: Ellipsoidal approximation of convex body, kls95
• Lemma 3.11: Lemma 3.2 in j21