In the classic Maximum Weight Independent Set problem we are given a graph $G$ with a nonnegative weight function on vertices, and the goal is to find an independent set in $G$ of maximum possible weight. While the problem is NP-hard in general, we give a polynomial-time algorithm working on any $P_6$-free graph, that is, a graph that has no path on $6$ vertices as an induced subgraph. This improves the polynomial-time algorithm on $P_5$-free graphs of Lokshtanov et al. (SODA 2014), and the quasipolynomial-time algorithm on $P_6$-free graphs of Lokshtanov et al (SODA 2016). The main technical contribution leading to our main result is enumeration of a polynomial-size family $\mathcal{F}$ of vertex subsets with the following property: for every maximal independent set $I$ in the graph, $\mathcal{F}$ contains all maximal cliques of some minimal chordal completion of $G$ that does not add any edge incident to a vertex of $I$.

中文翻译：

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无 $P_6$ 图形上最大权重独立集的多项式时间算法

在经典的最大权重独立集问题中，我们给出了一个在顶点上具有非负权重函数的图 $G$，目标是在 $G$ 中找到最大可能权重的独立集。虽然该问题通常是 NP-hard 问题，但我们给出了一个多项式时间算法，该算法适用于任何 $P_6$-free 图，即一个在 $6$ 顶点上没有路径的图作为诱导子图。这改进了 Lokshtanov 等人的 $P_5$-free 图上的多项式时间算法。(SODA 2014)，以及 Lokshtanov 等人 (SODA 2016) 的无 $P_6$ 图上的拟多项式时间算法。导致我们主要结果的主要技术贡献是枚举具有以下属性的顶点子集的多项式大小族 $\mathcal{F}$：对于图中的每个最大独立集 $I$，