Treewidth versus clique number. II. Tree-independence number

Abstract

In 2020, Dallard, Milanič, and Štorgel initiated a systematic study of graph classes in which the treewidth can only be large due the presence of a large clique, which they 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, an interesting open problem is whether (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 H-Packing problem, for any fixed finite set 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 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. We show that if a graph is given together with a tree decomposition with bounded independence number, then for any fixed finite set H of connected graphs, the Maximum Weight Independent H-Packing problem can be solved in polynomial time. Motivated by this result, we consider six graph containment relations-the subgraph, topological minor, and minor relations, as well as their induced variants-and for each of them characterize the graphs H for which any graph excluding H with respect to the relation admits a tree decomposition with bounded independence number. These results build on and refine the analogous characterizations due to Dallard, Milanič, and Štorgel for (tw, ω)-boundedness, and shows that in all these cases, (tw, ω)boundedness is actually equivalent to bounded tree-independence number. We use a variety of tools including SPQR trees and potential maximal cliques, and show that in the bounded cases, one can also obtain tree decompositions with bounded independence number efficiently. This 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...

Treewidth Versus Clique Number in Graph Classes with a Forbidden Structure

“…Besides the class of chordal graphs, further examples of (tw, ω)-bounded graph classes include the class of circular-arc graphs, as well as several more general families of graph classes studied in the literature (see [13, 15, 25-28, 44, 73, 74]). This is the third paper of a series of papers on (tw, ω)-bounded graph classes, initiated in [36] (see also [34]) and continued in [35].…”

“…It is known that (tw, ω)-bounded graph classes enjoy good algorithmic properties related to clique and coloring problems, in some cases under the mere assumption that the class admits a computable (tw, ω)-binding function (see [27,28,[34][35][36]). This motivates the question of whether (tw, ω)-boundedness has any other algorithmic implications, in particular, for problems related to independent sets.…”

“…This motivates the question of whether (tw, ω)-boundedness has any other algorithmic implications, in particular, for problems related to independent sets. A partial answer to this question was given in the second paper of the series [35] where we identified a sufficient condition for (tw, ω)-bounded graph classes to admit a polynomialtime algorithm for the Max Weight Independent Packing problem and, as a consequence, for the weighted variants of the Independent Set and Induced Matching problems. (We refer to Section 2 for precise definitions.)…”

“…The tree-independence number of a graph G is the minimum independence number over all tree decompositions of G. It follows from Ramsey's theorem that every graph class with bounded tree-independence number is (tw, ω)-bounded, even with a polynomial binding function; in particular, every graph class with bounded tree-independence number is polynomially χ-bounded. It was shown in [35] that the Max Weight Independent Packing problem can be solved in polynomial time provided that the input graph is given along with a tree decomposition with bounded independence number. Thus, boundedness of the tree-independence number is an algorithmically useful refinement of (tw, ω)-boundedness that still captures a wide variety of 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 (see [35] for details).…”

“…It was shown in [35] that the Max Weight Independent Packing problem can be solved in polynomial time provided that the input graph is given along with a tree decomposition with bounded independence number. Thus, boundedness of the tree-independence number is an algorithmically useful refinement of (tw, ω)-boundedness that still captures a wide variety of 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 (see [35] for details).…”