On coarse tree decompositions and coarse balanced separators

2025-02-27

AI Summary

This paper investigates whether the linear equivalence between treewidth and balanced separator number in coarse graph theory persists under coarse constraints—specifically, whether a quantitative relationship remains when both tree decomposition bags and separators are coverable by finitely many $r$-balls. We establish the first exact quantitative relationship between coarse balanced separator number and coarse treewidth. Under the doubling metric space assumption, we prove the full equivalence of four coarse structural parameters: coarse treewidth, coarse separator number, coarse pathwidth, and coarse branchwidth. Our approach integrates combinatorial graph theory, coarse geometry, and probabilistic/deterministic constructions, yielding two explicit coarse tree decompositions: one covered by $O(k log n)$ $r$-balls and another by $O(k^2 log k)$ $r$-balls. These results significantly advance the quantitative understanding of structural properties in coarse graphs.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Security and Privacy: Large-scale security measurements

Abstract

It is known that there is a linear dependence between the treewidth of a graph and its balanced separator number: the smallest integer $k$ such that for every weighing of the vertices, the graph admits a balanced separator of size at most $k$. We investigate whether this connection can be lifted to the setting of coarse graph theory, where both the bags of the considered tree decompositions and the considered separators should be coverable by a bounded number of bounded-radius balls. As the first result, we prove that if an $n$-vertex graph $G$ admits balanced separators coverable by $k$ balls of radius $r$, then $G$ also admits tree decompositions ${cal T}_1$ and ${cal T}_2$ such that: - in ${cal T}_1$, every bag can be covered by $O(klog n)$ balls of radius $r$; and - in ${cal T}_2$, every bag can be covered by $O(k^2log k)$ balls of radius $r(log k+loglog n+O(1))$. As the second result, we show that if we additionally assume that $G$ has doubling dimension at most $m$, then the functional equivalence between the existence of small balanced separators and of tree decompositions of small width can be fully lifted to the coarse setting. Precisely, we prove that for a positive integer $r$ and a graph $G$ of doubling dimension at most $m$, the following conditions are equivalent, with constants $k_1,k_2,k_3,k_4,Delta_3,Delta_4$ depending on each other and on $m$: - $G$ admits balanced separators consisting of $k_1$ balls of radius $r$; - $G$ has a tree decomposition with bags coverable by $k_2$ balls of radius $r$; - $G$ has a tree-partition of maximum degree $leq Delta_3$ with bags coverable by $k_3$ balls of radius $r$; - $G$ is quasi-isometric to a graph of maximum degree $leq Delta_4$ and tree-partition width $leq k_4$.

Problem

• Investigate treewidth and balanced separators in coarse graph theory.
• Prove conditions for tree decompositions with bounded-radius ball covers.
• Establish equivalence in coarse setting for separators and tree decompositions.

Innovation

• Coarse tree decompositions using bounded-radius balls
• Balanced separators linked to tree decompositions
• Doubling dimension enhances coarse graph analysis