整数最小值的凸最小化在 $\widetilde O(n^4)$ 时间内

通过鲁棒交替最小化在近线性时间内完成低秩矩阵补全

Low Rank Matrix Completion via Robust Alternating Minimization in Nearly Linear Time

本文提出一种对近似更新误差具有鲁棒性的交替最小化框架，将低秩矩阵补全的运行时间从近线性于验证时间的角度改进为 $\widetilde O(|Ω| k)$，优于此前 $\widetilde O(|Ω| k^2)$ 的方法。

Online Geometric Discrepancy for Stochastic Arrivals with Applications to Envy Minimization

Consider a unit interval $[0,1]$ in which $n$ points arrive one-by-one independently and uniformly at random. On arrival of a point, the problem is to immediately and irrevocably color it in $\{+1,-1\}$ while ensuring that every interval $[a,b] \subseteq [0,1]$ is nearly-balanced. We define \emph{discrepancy} as the largest imbalance of any interval during the entire process. If all the arriving points were known upfront then we can color them alternately to achieve a discrepancy of $1$. What is the minimum possible expected discrepancy when we color the points online? We show that the discrep…

Algorithms for Lipschitz Learning on Graphs

We develop fast algorithms for solving regression problems on graphs where one is given the value of a function at some vertices, and must find its smoothest possible extension to all vertices. The extension we compute is the absolutely minimal Lipschitz extension, and is the limit for large $p$ of $p$-Laplacian regularization. We present an algorithm that computes a minimal Lipschitz extension in expected linear time, and an algorithm that computes an absolutely minimal Lipschitz extension in expected time $\widetilde{O} (m n)$. The latter algorithm has variants that seem to run much faster i…

On the Efficiency of Entropic Regularized Algorithms for Optimal Transport

We present several new complexity results for the entropic regularized algorithms that approximately solve the optimal transport (OT) problem between two discrete probability measures with at most $n$ atoms. First, we improve the complexity bound of a greedy variant of Sinkhorn, known as \textit{Greenkhorn}, from $\widetilde{O}(n^2\varepsilon^{-3})$ to $\widetilde{O}(n^2\varepsilon^{-2})$. Notably, our result can match the best known complexity bound of Sinkhorn and help clarify why Greenkhorn significantly outperforms Sinkhorn in practice in terms of row/column updates as observed by~\citet{A…

对称子模最小化的比较模型

Symmetric Submodular Minimization from Comparisons

本文研究对称子模函数最小化的比较模型，提出确定性多项式时间算法，使用 O(n^3) 次比较找到非平凡最小化器，并解决加权最小割的开放问题，同时给出下界。

A Quantum Optimization Algorithm for Single Machine Total Weighted Tardiness Minimization

A single machine total weighted tardiness minimization (TWTM) problem in operational planning is considered. The problem is formulated as an NP-hard constrained combinatorial problem, which has no known deterministic polynomial complexity solution using classical computing. Based on efficient Grover's quantum search and Trugenberger's quantum optimization algorithms, a novel efficient quantum optimization algorithm is proposed to solve the NP-hard single machine TWTM problem, which makes the desired solution satisfying the searching constraints and showing the minimal TWT value be measured wit…