Solving Tall Dense Linear Programs in Nearly Linear Time

Solving SDP Faster: A Robust IPM Framework and Efficient Implementation

This paper introduces a new robust interior point method analysis for semidefinite programming (SDP). This new robust analysis can be combined with either logarithmic barrier or hybrid barrier. Under this new framework, we can improve the running time of semidefinite programming (SDP) with variable size $n \times n$ and $m$ constraints up to $ε$ accuracy. We show that for the case $m = Ω(n^2)$, we can solve SDPs in $m^ω$ time. This suggests solving SDP is nearly as fast as solving the linear system with equal number of variables and constraints. This is the first result that tall dense SDP can…

Bipartite Matching in Nearly-linear Time on Moderately Dense Graphs

We present an $\tilde O(m+n^{1.5})$-time randomized algorithm for maximum cardinality bipartite matching and related problems (e.g. transshipment, negative-weight shortest paths, and optimal transport) on $m$-edge, $n$-node graphs. For maximum cardinality bipartite matching on moderately dense graphs, i.e. $m = Ω(n^{1.5})$, our algorithm runs in time nearly linear in the input size and constitutes the first improvement over the classic $O(m\sqrt{n})$-time [Dinic 1970; Hopcroft-Karp 1971; Karzanov 1973] and $\tilde O(n^ω)$-time algorithms [Ibarra-Moran 1981; Mucha-Sankowski 2004] (where current…

Minimum Cost Flows, MDPs, and $\ell_1$-Regression in Nearly Linear Time for Dense Instances

In this paper we provide new randomized algorithms with improved runtimes for solving linear programs with two-sided constraints. In the special case of the minimum cost flow problem on $n$-vertex $m$-edge graphs with integer polynomially-bounded costs and capacities we obtain a randomized method which solves the problem in $\tilde{O}(m+n^{1.5})$ time. This improves upon the previous best runtime of $\tilde{O}(m\sqrt{n})$ (Lee-Sidford 2014) and, in the special case of unit-capacity maximum flow, improves upon the previous best runtimes of $m^{4/3+o(1)}$ (Liu-Sidford 2020, Kathuria 2020) and $\…

中等稠密图上近线性时间的广义流

Generalized Flow in Nearly-linear Time on Moderately Dense Graphs

本文针对带边损耗因子的有向图广义最大流与最小费用流问题，提出新的动态数据结构、谱结果和内点法随机算法，在中等稠密图上实现近线性时间。

Nested Dissection Meets IPMs: Planar Min-Cost Flow in Nearly-Linear Time

We present a nearly-linear time algorithm for finding a minimum-cost flow in planar graphs with polynomially bounded integer costs and capacities. The previous fastest algorithm for this problem is based on interior point methods (IPMs) and works for general sparse graphs in $O(n^{1.5}\text{poly}(\log n))$ time [Daitch-Spielman, STOC'08]. Intuitively, $Ω(n^{1.5})$ is a natural runtime barrier for IPM-based methods, since they require $\sqrt{n}$ iterations, each routing a possibly-dense electrical flow. To break this barrier, we develop a new implicit representation for flows based on generaliz…

近似二次时间的0-1背包问题

0-1 Knapsack in Nearly Quadratic Time

本文针对0-1背包问题提出确定性算法，将时间复杂度改进至O(n + w_max^2 log^4 w_max)，通过推广细粒度邻近性技术和扩展见证传播方法，弥合了上界与条件下界之间的差距。