A minimum chain cover (MCC) of a $k$-width directed acyclic graph (DAG) $G = (V, E)$ is a set of $k$ chains (paths in the transitive closure) of $G$ such that every vertex appears in at least one chain in the cover. The state-of-the-art solutions for MCC run in time $\tilde{O}(k(|V|+|E|))$ [M\"akinen et at., TALG], $O(T_{MF}(|E|) + k|V|)$, $O(k^2|V| + |E|)$ [C\'aceres et al., SODA 2022], $\tilde{O}(|V|^{3/2} + |E|)$ [Kogan and Parter, ICALP 2022] and $\tilde{O}(T_{MCF}(|E|) + \sqrt{k}|V|)$ [Kogan and Parter, SODA 2023], where $T_{MF}(|E|)$ and $T_{MCF}(|E|)$ are the running times for solving maximum flow (MF) and minimum-cost flow (MCF), respectively. In this work we present an algorithm running in time $O(T_{MF}(|E|) + (|V|+|E|)\log{k})$. By considering the recent result for solving MF [Chen et al., FOCS 2022] our algorithm is the first running in almost linear time. Moreover, our techniques are deterministic and derive a deterministic near-linear time algorithm for MCC if the same is provided for MF. At the core of our solution we use a modified version of the mergeable dictionaries [Farach and Thorup, Algorithmica], [Iacono and \"Ozkan, ICALP 2010] data structure boosted with the SIZE-SPLIT operation and answering queries in amortized logarithmic time, which can be of independent interest.

翻译：一个$k$宽度有向无环图（DAG）$G = (V, E)$的最小链覆盖（MCC）是指$G$的一组$k$条链（传递闭包中的路径），使得每个顶点至少出现在覆盖中的一条链中。当前最先进的MCC解决方案的运行时间为$\tilde{O}(k(|V|+|E|))$ [M\"akinen 等人，TALG]，$O(T_{MF}(|E|) + k|V|)$，$O(k^2|V| + |E|)$ [C\'aceres 等人，SODA 2022]，$\tilde{O}(|V|^{3/2} + |E|)$ [Kogan 和 Parter，ICALP 2022] 以及$\tilde{O}(T_{MCF}(|E|) + \sqrt{k}|V|)$ [Kogan 和 Parter，SODA 2023]，其中$T_{MF}(|E|)$和$T_{MCF}(|E|)$分别是求解最大流（MF）和最小费用流（MCF）的运行时间。在本文中，我们提出了一种运行时间为$O(T_{MF}(|E|) + (|V|+|E|)\log{k})$的算法。结合求解MF的最新成果 [Chen 等人，FOCS 2022]，我们的算法是首个在近乎线性时间内运行的算法。此外，我们的技术是确定性的，并且如果MF算法是确定性的，则可推导出确定性的近线性时间MCC算法。我们解决方案的核心是使用了合并字典 [Farach 和 Thorup，Algorithmica]，[Iacono 和 \"Ozkan，ICALP 2010] 数据结构的改进版本，该版本通过SIZE-SPLIT操作增强，并在摊销对数时间内回答查询，这本身可能具有独立的研究价值。