We study the tractability of the maximum independent set problem from the viewpoint of graph width parameters, with the goal of defining a width parameter that is as general as possible and allows to solve independent set in polynomial-time on graphs where the parameter is bounded. We introduce two new graph width parameters: one-sided maximum induced matching-width (o-mim-width) and neighbor-depth. O-mim-width is a graph parameter that is more general than the known parameters mim-width and tree-independence number, and we show that independent set and feedback vertex set can be solved in polynomial-time given a decomposition with bounded o-mim-width. O-mim-width is the first width parameter that gives a common generalization of chordal graphs and graphs of bounded clique-width in terms of tractability of these problems. The parameter o-mim-width, as well as the related parameters mim-width and sim-width, have the limitation that no algorithms are known to compute bounded-width decompositions in polynomial-time. To partially resolve this limitation, we introduce the parameter neighbor-depth. We show that given a graph of neighbor-depth $k$, independent set can be solved in time $n^{O(k)}$ even without knowing a corresponding decomposition. We also show that neighbor-depth is bounded by a polylogarithmic function on the number of vertices on large classes of graphs, including graphs of bounded o-mim-width, and more generally graphs of bounded sim-width, giving a quasipolynomial-time algorithm for independent set on these graph classes. This resolves an open problem asked by Kang, Kwon, Str{\o}mme, and Telle [TCS 2017].

翻译：我们从图宽度参数的角度研究最大独立集问题的可追踪性，目标是定义一种尽可能通用的宽度参数，使得当参数有界时，可在多项式时间内求解独立集问题。我们引入两种新的图宽度参数：单侧最大诱导匹配宽度（o-mim-width）和邻域深度（neighbor-depth）。o-mim-width是一个比已知参数mim-width和树独立数（tree-independence number）更通用的图参数，我们证明在给定有界o-mim-width分解的情况下，独立集问题和反馈顶点集问题可在多项式时间内求解。o-mim-width是首个在问题可追踪性方面统一概括弦图（chordal graphs）和有界团宽度（clique-width）图的宽度参数。参数o-mim-width及相关参数mim-width和sim-width均存在局限性：目前尚无已知算法能在多项式时间内计算有界宽度的分解。为部分解决该问题，我们引入邻域深度参数。我们证明，对于邻域深度为k的图，即使不知道相应的分解，独立集问题也可在n^{O(k)}时间内求解。我们还证明，在包含有界o-mim-width图及更一般的有界sim-width图在内的众多图类中，邻域深度关于顶点数具有多对数函数界，从而为这些图类上的独立集问题提供了拟多项式时间算法。这解决了Kang、Kwon、Strømme和Telle [TCS 2017]提出的开放问题。