SIAM Journal on Discrete Mathematics, Volume 34, Issue 2, Page 1385-1408, January 2020.
We investigate the odd multiway node (edge) cut problem where the input is a graph with a specified collection of terminal nodes, and the goal is to find a smallest subset of non-terminal nodes (edges) to delete so that the terminal nodes do not have an odd length path between them. In an earlier work, Lokshtanov and Ramanujan showed that both odd multiway node cut and odd multiway edge cut are fixed-parameter tractable (FPT) when parameterized by the size of the solution in undirected graphs. In this work, we focus on directed acyclic graphs (DAGs) and design a fixed-parameter algorithm. Our main contribution is a broadening of the shadow-removal framework to address parity problems in DAGs. We complement our FPT results with tight approximability as well as polyhedral results for two terminals in DAGs. Additionally, we show inapproximability results for odd multiway edge cut in undirected graphs even for two terminals.

中文翻译：

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有向无环图的奇数多路割

SIAM离散数学杂志，第34卷，第2期，第1385-1408页，2020年1月。
我们研究奇数多路节点（边缘）切问题，其中输入是具有指定终端节点集合的图，并且目标是找到要删除的最小非终端节点（边缘）子集，以便终端节点执行它们之间没有奇怪的长度路径。在更早的工作中，Lokshtanov和Ramanujan表明，在无向图中通过解的大小进行参数化时，奇数多路节点切割和奇数多路边缘切割都是固定参数可处理的（FPT）。在这项工作中，我们专注于有向无环图（DAG）并设计固定参数算法。我们的主要贡献是扩大了阴影去除框架，以解决DAG中的奇偶校验问题。我们用DAG中两个终端的紧密近似性以及多面体结果对FPT结果进行补充。另外，