发表机构
Université Libre de Bruxelles; Universität Kassel（布鲁塞尔自由大学; 卡塞尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一个简单算法，在独立集多面体上以摊还延迟O(n)计算哈密顿路径，并推广到匹配多面体、链多面体和序多面体，通过贪心规则绕过极大权重独立集难题。
AI 中文摘要
图$G$的独立集多面体（或称稳定集多面体）是由$G$的所有独立集的特征向量的凸包定义的0/1多面体。我们提出了一个简单算法，用于在给定$n$顶点图$G$的独立集多面体上计算哈密顿路径，其摊还延迟为$\mathcal{O}(n)$。独立集的列出方式使得两个连续集合的差异要么是移除一个顶点，要么是添加一个顶点并同时从独立集中移除其邻居，即两个连续独立集之间的对称差在$G$中诱导出一个星形。作为该结果的应用，我们获得了一个算法，用于在$m$边图$G$的匹配多面体上计算哈密顿路径，其最坏情况延迟为$\mathcal{O}(m)$，该算法以这样的方式列出$G$的所有匹配，使得两个连续匹配之间的对称差是一条至多三条边的路径。此外，我们获得了一个算法，用于在$n$元素偏序集$P$的链多面体和序多面体上计算哈密顿路径，其摊还延迟为$\mathcal{O}(n)$，该算法分别通过星形交换列出$P$的所有反链或所有理想。我们的算法源自Merino和Mütze（FOCS'23+SICOMP'24）提出的用于在任意0/1多面体上计算哈密顿路径的通用框架，该框架使用线性优化过程作为黑盒。我们的算法通过一个简单且纯组合的贪心规则绕过了计算上难以处理的极大权重独立集问题。
英文摘要
The independent set polytope, or stable set polytope, of a graph $G$ is the 0/1-polytope defined by the convex hull of the characteristic vectors of all independent sets of $G$. We present a simple algorithm for computing a Hamilton path on the independent set polytope of a given $n$-vertex graph $G$ with amortized delay $\mathcal{O}(n)$. The independent sets are listed such that two consecutive sets differ either in removing a vertex, or adding a vertex and removing its neighbors from the independent set, i.e., the symmetric difference between two consecutive independent sets induces a star in $G$. As applications of this result, we obtain an algorithm to compute a Hamilton path on the matching polytope of an $m$-edge graph $G$ with worst-case delay $\mathcal{O}(m)$, which lists all matchings of $G$ in such a way that the symmetric difference between two consecutive matchings is a path on at most three edges. Furthermore, we obtain an algorithm to compute a Hamilton path on the chain polytope and order polytope of an $n$-element poset $P$ with amortized delay $\mathcal{O}(n)$, which lists all antichains of $P$ or all ideals of $P$, respectively, by star exchanges. Our algorithms are derived from the generic framework proposed by Merino and Mütze (FOCS'23+SICOMP'24) for computing Hamilton paths on arbitrary 0/1-polytopes, which uses a linear optimization procedure as a black box. Our algorithms bypass solving the computationally intractable maximum weight independent set problem by a simple and purely combinatorial greedy rule.
