Tight Algorithmic Applications of Clique-Width Generalizations

TL;DR

This paper explores two generalizations of clique-width, multi-clique-width and fusion-width, establishing their relationships and demonstrating that many problems remain efficiently solvable when parameterized by these measures, despite their differences from clique-width.

Contribution

The paper establishes the relation between multi-clique-width and fusion-width, and shows that problems solvable with clique-width can also be tackled efficiently using these generalized parameters.

Findings

Multi-clique-width is at most fusion-width plus one.

Problems like Connected Dominating Set are solvable in similar time using multi-clique-width as with clique-width.

For some problems, fusion-width allows algorithms with comparable efficiency, supported by new glue-expression techniques.

Abstract

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.