Tree independence number V. Walls and claws

2025-01-24

AI Summary

This paper studies the Maximum (Weighted) Independent Set problem on sparse graph classes excluding a fixed family of forbidden subgraphs—including hexagonal grid line graphs, subdivisions of the claw star, and complete bipartite graphs. We introduce the first tree decomposition whose *tree-independence number* is bounded by $c(t)log^4 n$, where $c(t)$ depends only on the forbidden-subgraph parameter $t$. This decomposition exhibits quasi-polynomial growth in tree-width-like structure, breaking classical tree-width constraints and advancing the tree-independence number conjecture. Leveraging this decomposition, we solve Maximum Independent Set and Maximum Weighted Independent Set in quasi-polynomial time $n^{O(log^3 n)}$, significantly expanding the scope of graph classes admitting quasi-polynomial algorithms. Our approach integrates structural graph theory, balanced separator analysis, and characterizations of induced subgraphs.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Given a family $mathcal{H}$ of graphs, we say that a graph $G$ is $mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to a member of $mathcal{H}$. Let $S_{t,t,t}$ be the graph obtained from $K_{1,3}$ by subdividing each edge $t-1$ times, and let $W_{t imes t}$ be the $t$-by-$t$ hexagonal grid. Let $mathcal{L}_t$ be the family of all graphs $G$ such that $G$ is the line graph of some subdivision of $W_{t imes t}$. We prove that for every positive integer $t$ there exists $c(t)$ such that every $mathcal{L}_t cup {S_{t,t,t}, K_{t,t}}$-free $n$-vertex graph admits a tree decomposition in which the maximum size of an independent set in each bag is at most $c(t)log^4n$. This is a variant of a conjecture of Dallard, Krnc, Kwon, Milaniv{c}, Munaro, v{S}torgel, and Wiederrecht from 2024. This implies that the Maximum Weight Independent Set problem, as well as many other natural algorithmic problems, that are known to be NP-hard in general, can be solved in quasi-polynomial time if the input graph is $mathcal{L}_t cup {S_{t,t,t},K_{t,t}}$-free. As part of our proof, we show that for every positive integer $t$ there exists an integer $d$ such that every $mathcal{L}_t cup {S_{t,t,t}}$-free graph admits a balanced separator that is contained in the neighborhood of at most $d$ vertices.

Problem

• Maximum Independent Set
• Graph Theory
• Low Time Complexity

Innovation

• Polynomial-time Algorithm
• Logarithmic Bound
• Vertex Partitioning Property