更快更简单地遍历0/1多面体
Faster and simpler traversal of 0/1-polytopes
中文总结 AI 辅助
本文改进了在0/1多面体骨架上计算哈密顿路径的算法，使其更简单快速，摊还延迟仅比优化算法运行时间大常数倍，还得到相关问题改进算法及0/1多面体顶点枚举的更快算法。
中文摘要 AI 辅助
最近，Merino和Mütze（FOCS'23+SICOMP'24）提出了一种算法，用于在任何0/1多面体${\rm conv}(X)$（其中$X\subseteq\{0,1\}^n$）的骨架上计算哈密顿路径。该算法将求解经典线性优化问题$\min\{w\cdot x\mid x\in X\}$（对于某个权重向量$w\in\mathbb{R}^n$）的算法作为黑盒使用。所得哈密顿路径上每个访问顶点的延迟仅比求解一个优化算法实例的时间大$\log n$倍。在本文中，我们使哈密顿路径算法更简单更快。即，我们获得了一个摊还延迟，仅比优化算法的运行时间大一个常数倍，从而消除了$\log n$因子。具体结果是，这产生了用于生成拟阵中的基和独立集、图中的生成树、森林、匹配和最大匹配、顶点覆盖、最小顶点覆盖、二分图中的独立集和最大独立集以及偏序集中的反链、最大反链和理想的改进算法。所有这些列表都对应于相应多面体上的哈密顿路径。此外，我们为0/1多面体$\{x\in\mathbb{R}^n\mid Ax\leq b\}$（其中$A\in \mathbb{R}^{m\times n}$且$b\in\mathbb{R}^m$）的顶点枚举问题获得了一个$\mathcal{O}(t_{\rm LP})$摊还延迟算法，其中$t_{\rm LP}$是求解线性规划$\min\{w\cdot x\mid Ax\leq b\}$所需的时间。这改进了Merino和Mütze的$\mathcal{O}(t_{\rm LP} \log n)$延迟算法以及1998年Bussieck和Lübbecke的先前$\mathcal{O}(t_{\rm LP}\,n)$延迟算法。
英文摘要
Recently, Merino and Mütze (FOCS'23+SICOMP'24) presented an algorithm for computing a Hamilton path on the skeleton of any 0/1-polytope ${\rm conv}(X)$, where $X\subseteq\{0,1\}^n$. The algorithm uses as a black box an algorithm for solving the classical linear optimization problem $\min\{w\cdot x\mid x\in X\}$ for some weight vector $w\in\mathbb{R}^n$. The resulting delay per visited vertex on the Hamilton path is only by a $\log n$ factor larger than the time to solve one instance of the optimization algorithm. In this paper, we make the Hamilton path algorithm simpler and faster. Namely, we obtain an amortized delay that is only by a constant factor larger than the running time of the optimization algorithm, thus removing the $\log n$ factor. As concrete results, this yields improved algorithms for generating bases and independent sets in a matroid, spanning trees, forests, matchings and maximum matchings in a graph, vertex covers, minimum vertex covers, independent sets and maximum independent sets in a bipartite graph, and antichains, maximum antichains and ideals in a poset. All of these listings correspond to Hamilton paths on the corresponding polytopes. Furthermore, we obtain an $\mathcal{O}(t_{\rm LP})$ amortized delay algorithm for the vertex enumeration problem on 0/1-polytopes $\{x\in\mathbb{R}^n\mid Ax\leq b\}$, where $A\in \mathbb{R}^{m\times n}$ and $b\in\mathbb{R}^m$, and $t_{\rm LP}$ is the time needed to solve the linear program $\min\{w\cdot x\mid Ax\leq b\}$. This improves upon the $\mathcal{O}(t_{\rm LP} \log n)$ delay algorithm of Merino and Mütze, and the previous $\mathcal{O}(t_{\rm LP}\,n)$ delay algorithm of Bussieck and Lübbecke from 1998.
