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解性，旨在定义一种尽可能通用的宽度参数，使得当该参数有界时，可以在多项式时间内求解独立集问题。我们引入了两种新的图宽度参数：单侧最大诱导匹配宽度（o-mim-width）和邻居深度。O-mim-width是一种比已知参数mim-width和树独立数更通用的图参数，我们证明在给定具有有界o-mim-width的分解时，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是首个在弦图和有界团宽度的图上统一这些问题的可解性的宽度参数。参数o-mim-width及其相关参数mim-width和sim-width存在局限性：目前尚无算法能在多项式时间内计算有界宽度的分解。为部分解决这一局限，我们引入了邻居深度参数。我们证明，给定一个邻居深度为$k$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，邻居深度受顶点数的多对数函数界限制，这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]提出的一个开放问题。