多分离器多面体的典范面
The canonical facets of multi-separator polytopes
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中文总结 AI 辅助
本研究针对Irmai等人2024年提出的图多分离器问题开展多面体研究，刻画其整数线性规划不等式诱导的面，得到路径情形下的完全对偶整数描述，关联布尔二次多面体与提升多割多面体。
中文摘要 AI 辅助
本研究针对Irmai等人（2024）提出的图多分离器问题开展多面体研究，该问题作为提升多割问题的替代方案应用于图像分割任务。从整数线性规划（ILP）公式及其可行解张成的多分离器多面体出发，我们依据可高效判定的图论条件刻画了ILP不等式诱导的所有面。接着，我们对这些不等式进行强化，并描述了部分多分离器多面体由强化不等式诱导的额外面。具体而言，我们在所有顶点对均需分离的路径情形下，得到了多分离器多面体的完全对偶整数描述。最后，我们将多分离器多面体与布尔二次多面体关联，证明奇圈不等式诱导的面无法通用传递；还将其与提升多割多面体关联，表明两个多面体互为对方某一面的投影。
英文摘要
We initiate a polyhedral study of the graph multi-separator problem proposed by Irmai et al. (2024) as an alternative to the lifted multicut problem for application to the task of image segmentation. Starting with an integer linear program (ILP) formulation and the multi-separator polytope spanned by its feasible solutions, we characterize in terms of efficiently-decidable, graph-theoretic conditions all facets induced by inequalities of the ILP. We proceed by strengthening these inequalities and describing additional facets of some multi-separator polytopes induced by the stronger inequalities. Specifically, we obtain a totally dual integral description of the multi-separator polytope for paths in the case where separation is considered for all vertex pairs. Finally, we relate the multi-separator polytope to the boolean quadric polytope, showing that facets induced by odd-cycle inequalities do not transfer generally, and to the lifted multicut polytope, showing that either polytope is a projection of a face of the other.
发表机构
- Faculty of Computer Science, TU Dresden（德累斯顿工业大学计算机学院）
- Center for Scalable Data Analytics and AI (ScaDS.AI) Dresden/Leipzig（德累斯顿/莱ipzig可扩展数据分析与人工智能中心）
- Université Sorbonne Paris Nord（巴黎北索邦大学）
- CNRS（法国国家科学研究中心）
机构由 AI 辅助整理，请以论文原文为准。
