In this work, we study two natural generalizations of clique-width introduced by Martin F\"urer. Multi-clique-width (mcw) allows every vertex to hold multiple labels [ITCS 2017], while for fusion-width (fw) we have a possibility to merge all vertices of a certain label [LATIN 2014]. F\"urer has shown that both parameters are upper-bounded by treewidth thus making them more appealing from an algorithmic perspective than clique-width and asked for applications of these parameters for problem solving. First, we determine the relation between these two parameters by showing that $\operatorname{mcw} \leq \operatorname{fw} + 1$. Then we show that when parameterized by multi-clique-width, many problems (e.g., Connected Dominating Set) admit algorithms with the same running time as for clique-width despite the exponential gap between these two parameters. For some problems (e.g., Hamiltonian Cycle) we show an analogous result for fusion-width: For this we present an alternative view on fusion-width by introducing so-called glue-expressions which might be interesting on their own. All algorithms obtained in this work are tight up to (Strong) Exponential Time Hypothesis.

翻译：本文研究了Martin Fürer引入的两种团宽度自然推广。多团宽度（mcw）允许每个顶点持有多个标签[ITCS 2017]，而融合宽度（fw）则允许合并具有特定标签的所有顶点[LATIN 2014]。Fürer已证明这两个参数均受树宽上界控制，因此从算法视角比团宽度更具吸引力，并提出了这些参数在问题求解中的应用需求。首先，我们通过证明$\operatorname{mcw} \leq \operatorname{fw} + 1$确定了这两个参数之间的关系。其次，我们表明当以多团宽度为参数时，许多问题（如连通支配集）可采用与团宽度相同运行时间的算法，尽管这两个参数之间存在指数级差距。对于某些问题（如哈密顿环），我们展示了融合宽度的类似结果：为此，我们通过引入可能具有独立研究价值的所谓胶合表达式（glue-expressions），提供了融合宽度的一种新视角。本文获得的所有算法在（强）指数时间假设下均为紧致的。