多项式时间中无（$P_7$，Triangle）图的最大权独立集
Maximum Weight Independent Sets for ($P_7$,Triangle)-Free Graphs in Polynomial Time
摘要：
有限无向图的最大权独立集（MWIS）问题要求一组两两不相邻的顶点的最大权和。MWIS是研究最多和最重要的算法图问题之一;众所周知，它是NP-完全的，即使在各种强限制下，如无三角形图，它仍然是NP-完全的。它的复杂性是无$P_k$图$k \ge 5$的一个公开问题.最近，Lokshtanov，Vatshelle和Villanger证明了对于无$P_5$的图，MWIS可以在多项式时间内求解; Lokshtanov，Pilipczuk和van Leeuwen证明了对于无$P_6$的图，MWIS可以在准多项式时间内求解.对于无$P_k$的图$k \geq 6$，MWIS是否能在多项式时间内求解，或者对于无$P_k$的图$k \geq 7$，MWIS是否能在准多项式时间内求解，仍然是一个悬而未决的问题. $P_k$ -free图的一些刻画和一些进展在文献中已知，但到目前为止还没有解决这个问题。在本文中，我们表明，MWIS可以解决在多项式时间（$P_7$，三角形）-自由图。这推广了（$P_6$，triangle）-free图的相应结果，并可能为$P_7$ -free图的MWIS研究提供一些进展.
英文摘要：
The Maximum Weight Independent Set (MWIS) problem on finite undirected graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum weight sum. MWIS is one of the most investigated and most important algorithmic graph problems; it is well known to be NP-complete, and it remains NP-complete even under various strong restrictions such as for triangle-free graphs. Its complexity was an open problem for $P_k$-free graphs, $k \ge 5$. Recently, Lokshtanov, Vatshelle, and Villanger proved that MWIS can be solved in polynomial time for $P_5$-free graphs, and Lokshtanov, Pilipczuk, and van Leeuwen proved that MWIS can be solved in quasi-polynomial time for $P_6$-free graphs. It still remains an open problem whether MWIS can be solved in polynomial time for $P_k$-free graphs, $k \geq 6$ or in quasi-polynomial time for $P_k$-free graphs, $k \geq 7$. Some characterizations of $P_k$-free graphs and some progress are known in the literature but so far did not solve the problem. In this paper, we show that MWIS can be solved in polynomial time for ($P_7$,triangle)-free graphs. This extends the corresponding result for ($P_6$,triangle)-free graphs and may provide some progress in the study of MWIS for $P_7$-free graphs.
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