独立集的新宽度参数：单侧mim-width与邻居深度
New Width Parameters for Independent Set: One-sided-mim-width and Neighbor-depth
AI总结：
引入了两个新的图宽度参数o-mim-width和neighbor-depth，证明了在它们有界时独立集问题可在多项式或拟多项式时间内求解，并解决了相关开放问题。
摘要：
我们从图宽度参数的角度研究了最大独立集问题的可处理性，目标是定义一个尽可能一般的宽度参数，使得在参数有界的图上能够以多项式时间求解独立集。我们引入了两个新的图宽度参数：单侧最大诱导匹配宽度（o-mim-width）和邻居深度。o-mim-width是一个比已知参数mim-width和tree-independence number更一般的图参数，我们证明了在给定具有有界o-mim-width的分解时，独立集和反馈顶点集可以在多项式时间内求解。o-mim-width是首个在这些问题可处理性方面对弦图和有界clique-width图进行共同推广的宽度参数。\n 参数o-mim-width及相关参数mim-width和sim-width存在一个局限性，即目前没有已知算法能够在多项式时间内计算出有界宽度的分解。为了部分解决这一局限性，我们引入了参数邻居深度。我们证明了给定邻居深度为$k$的图，即使不知道相应的分解，独立集也可以在$n^{O(k)}$时间内求解。我们还证明了在大量图类上，包括有界o-mim-width的图，以及更一般的有界sim-width的图，邻居深度受限于顶点数量的多重对数函数，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]提出的一个开放问题。
英文摘要：
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, Strømme, and Telle [TCS 2017].
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
