Treewidth versus clique number. II. Tree-independence number

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-11-09 DOI:10.1016/j.jctb.2023.10.006

Clément Dallard , Martin Milanič , Kenny Štorgel

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引用次数: 13

Abstract

In 2020, we initiated a systematic study of graph classes in which the treewidth can only be large due to the presence of a large clique, which we call $(tw, ω)$ -bounded. The family of $(tw, ω)$ -bounded graph classes provides a unifying framework for a variety of very different families of graph classes, including graph classes of bounded treewidth, graph classes of bounded independence number, intersection graphs of connected subgraphs of graphs with bounded treewidth, and graphs in which all minimal separators are of bounded size. While Chaplick and Zeman showed in 2017 that $(tw, ω)$ -bounded graph classes enjoy some good algorithmic properties related to clique and coloring problems, it is an interesting open problem to which extent $(tw, ω)$ -boundedness has useful algorithmic implications for problems related to independent sets. We provide a partial answer to this question by identifying a sufficient condition for $(tw, ω)$ -bounded graph classes to admit a polynomial-time algorithm for the Maximum Weight Independent Packing problem and, as a consequence, for the weighted variants of the Independent Set and Induced Matching problems.

Our approach is based on a new min-max graph parameter related to tree decompositions. We define the independence number of a tree decomposition $T$ of a graph as the maximum independence number over all subgraphs of G induced by some bag of $T$ . The tree-independence number of a graph G is then defined as the minimum independence number over all tree decompositions of G. Boundedness of the tree-independence number is a refinement of $(tw, ω)$ -boundedness that is still general enough to hold for all the aforementioned families of graph classes. Generalizing a result on chordal graphs due to Cameron and Hell from 2006, we show that if a graph is given together with a tree decomposition with bounded independence number, then the Maximum Weight Independent Packing problem can be solved in polynomial time. Applications of our general algorithmic result to specific graph classes are given in the third paper of the series [Dallard, Milanič, and Štorgel, Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure].

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树宽与派系数。2Tree-independence数量

在2020年，我们开始了对图类的系统研究，其中树宽度只能由于存在一个大团而大，我们称之为(tw，ω)有界。(tw，ω)有界图类族为各种非常不同的图类族提供了一个统一的框架，包括有界树宽的图类，有界独立数的图类，有界树宽的图的连通子图的相交图，以及所有最小分隔符都有界大小的图。虽然Chaplick和Zeman在2017年表明(tw，ω)有界图类具有与团和着色问题相关的一些良好算法性质，但在某种程度上(tw，ω)有界性对与独立集相关的问题具有有用的算法含义，这是一个有趣的开放问题。我们通过确定(tw，ω)有界图类允许多项式时间算法用于最大权重独立填充问题的充分条件，从而为独立集和诱导匹配问题的加权变体提供了部分答案。我们的方法是基于与树分解相关的一个新的最小-最大图参数。我们定义独立的树分解T图的最大独立号码/ G的所有子图由一些袋T图G的tree-independence数量就可以定义为独立号码最小树分解的G . tree-independence数量是一个精致的有界性(tw,ω)有界性,仍然是一般足以容纳所有上述家庭的图类。推广了Cameron和Hell(2006)关于弦图的一个结果，证明了如果一个图与一个有界独立数的树分解一起给出，那么最大权无关填充问题可以在多项式时间内得到解决。我们的一般算法结果在特定图类中的应用在该系列的第三篇论文[Dallard, milaninik, and Štorgel, Treewidth vs . clique number]中给出。3具有禁止结构的图的树无关数]。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.