(Treewidth, Clique)-Boundedness and Poly-logarithmic Tree-Independence
Maria Chudnovsky, Ajaykrishnan E S, Daniel Lokshtanov
TL;DR
This work studies graph classes with poly-logarithmic tree-independence numbers, introducing independence-containers to generalize maximal cliques and linking TI-boundedness to traditional treewidth via a trio of equivalent conditions. It develops container-family techniques and separator-rounding machinery, including fractional and dual formulations for $A$--$B$ and balanced separators, to prove that poly-logarithmic $tw_{\alpha}(G)$ is equivalent to having treewidth bounded by a poly-logarithm of the clique number, for hereditary classes. The authors show that excluding certain induced subgraphs $\overline{kK_k}$ is precisely what enables quasi-polynomial container families and efficient separation, while providing obstructions and dual results that underpin the main equivalences. The framework yields quasi-polynomial-time consequences for problems like Independent Set on classes with TI-number bounds and opens several open problems about finer characterizations and optimal container sizes, connecting Ramsey-type phenomena, container methods, and graph decompositions to a unified poly-logarithmic regime.
Abstract
An {\em independent set} in a graph $G$ is a set of pairwise non-adjacent vertices. A {\em tree decomposition} of $G$ is a pair $(T, χ)$ where $T$ is a tree and $χ: V(T) \rightarrow 2^{V(G)}$ is a function satisfying the following two axioms: for every edge $uv \in V(G)$ there is a $x \in V(T)$ such that $\{u,v\} \subseteq χ(x)$, and for every vertex $u \in V(G)$ the set $\{x \in V(T) ~:~ u \in χ(X)\}$ induces a non-empty and connected subtree of $T$. The sets $χ(x)$ for $x \in V(T)$ are called the {\em bags} of the tree decomposition. The {\em tree-independence} number of $G$ is the minimum taken over all tree decompositions of $G$ of the size of the maximum independent set of the graph induced by a bag of the tree decomposition. The study of graph classes with bounded tree-independence number has attracted much attention in recent years, in part due its improtant algorithmic implications. A conjecture of Dallard, Milanič and Storgel, connecting tree-independence number to the classical notion of treewidth, was one of the motivating problems in the area. This conjecture was recently disproved, but here we prove a slight variant of it, that retains much of the algorithmic significance. As part of the proof we introduce the notion of {\em independence-containers}, which can be viewed as a generalization of the set of all maximal cliques of a graph, and is of independent interest.