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On parameterized algorithms for fixed-order book thickness with respect to the pathwidth of the vertex ordering
Theoretical Computer Science ( IF 0.9 ) Pub Date : 2021-04-27 , DOI: 10.1016/j.tcs.2021.04.021
Yunlong Liu , Jie Chen , Jingui Huang , Jianxin Wang

Given a graph G=(V,E) with a vertex ordering ≺, the fixed-order book thickness problem asks whether there is a page assignment σ such that ,σ is a k-page book embedding of G. This problem is NP-complete even for any fixed k greater than 3. Recently, Bhore et al.(GD 2019; J. Graph Algorithms Appl. 2020) presented a parameterized algorithm with respect to the pathwidth κ of the vertex ordering. In this paper, we first re-analyze the running time for Bhore et al.'s algorithm, and prove a bound of 2O(κ2)|V| improving Bhore et al.'s bound of κO(κ2)|V|. Then, we show that fixed-order book thickness parameterized by the pathwidth of the vertex ordering does not admit a polynomial kernel unless NP ⊆ coNP/poly. Finally, we show that a generalized fixed-order book thickness problem, in which a budget of at most c crossings over all pages was given, admits a parameterized algorithm running in time (c+2)O(κ2)|V|.



中文翻译:

关于定序书本厚度相对于顶点定序路径宽度的参数化算法

给定一个图 G=伏特E在顶点顺序为≺的情况下,固定顺序的书本厚度问题询问是否存在页面分配σ,使得σ是嵌入Gk页图书。即使对于任何大于3的固定k,这个问题也是NP完全的。最近,Bhore等人(GD 2019; J。Graph Algorithms Appl。2020)提出了一种关于顶点排序的路径宽度κ的参数化算法。在本文中,我们首先重新分析了Bhore等人算法的运行时间,并证明了2个Øκ2个|伏特| 改善Bhore等人的界限 κØκ2个|伏特|。然后,我们表明,除非NP⊆coNP / poly,否则由顶点排序的路径宽度参数化的固定顺序的书本厚度不容许多项式核。最后,我们证明了一个广义的固定顺序书本厚度问题,其中给出了所有页面上最多c个交叉点的预算,它接受了及时运行的参数化算法C+2个Øκ2个|伏特|

更新日期:2021-04-28
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