Tree decompositions with bounded independence number and their algorithmic applications

In 2020, Dallard, Milani\v{c}, and \v{S}torgel initiated a systematic study of graphs classes in which the treewidth can only be large due the presence of a large clique, which they call $(\textrm{tw},\omega)$-bounded. The family of $(\textrm{tw},\omega)$-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 $(\textrm{tw},\omega)$-bounded graph classes enjoy some good algorithmic properties related to clique and coloring problems, an interesting open problem is whether $(\textrm{tw},\omega)$-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 $(\textrm{tw},\omega)$-bounded graph classes to admit a polynomial-time algorithm for the Maximum Weight Independent $\mathcal{H}$-Packing problem, for any fixed finite set $\mathcal{H}$ of connected graphs. This family of problems generalizes several other problems studied in the literature, including the Maximum Weight Independent Set and Maximum Weight Induced Matching problems. Our approach leads to polynomial-time algorithms for the Maximum Weight Independent Set problem in an infinite family of graph classes, each of which properly contains the class of chordal graphs. These results also apply to the class of $1$-perfectly orientable graphs, answering a question of Beisegel, Chudnovsky, Gurvich, Milani\v{c}, and Servatius from 2019.