Generalized Graph Packing Problems Parameterized by Treewidth

2025-09-07

AI Summary

This paper investigates the generalized graph packing problem on graphs of bounded treewidth, encompassing vertex-disjoint H-packing and H-deletion, along with two natural generalizations: (1) vertices may be used at most c times, and (2) H is an arbitrary fixed graph—not necessarily a clique. Employing tree-decomposition-based dynamic programming, conditional lower bounds under ETH and SETH, and combinatorial structural analysis, we establish the first tight upper bound of $(c+1)^{mathrm{tw}} n^{O(1)}$ for c-restricted $K_d$-packing. Moreover, we prove that if some biconnected component of H is not a clique, then H-deletion admits no algorithm running in time $2^{o(mathrm{tw} log mathrm{tw})}$. These results precisely characterize the optimal time complexity of such problems parameterized by treewidth and reveal a fundamental complexity-theoretic transition—from tractability under clique constraints to inherent hardness for general graphs.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

$H$-Packing is the problem of finding a maximum number of vertex-disjoint copies of $H$ in a given graph $G$. $H$-Partition is the special case of finding a set of vertex-disjoint copies that cover each vertex of $G$ exactly once. Our goal is to study these problems and some generalizations on bounded-treewidth graphs. The case of $H$ being a triangle is well understood: given a tree decomposition of $G$ having treewidth $tw$, the $K_3$-Packing problem can be solved in time $2^{tw} cdot n^{O(1)}$, while Lokshtanov et al.~[{it ACM Transactions on Algorithms} 2018] showed, under the Strong Exponential-Time Hypothesis (SETH), that there is no $(2-ε)^{tw}cdot n^{O(1)}$ algorithm for any $ε>0$ even for $K_3$-Partition. Similar results can be obtained for any other clique $K_d$ for $dge 3$. We provide generalizations in two directions: - We consider a generalization of the problem where every vertex can be used at most $c$ times for some $cge 1$. When $H$ is any clique $K_d$ with $dge 3$, then we give upper and lower bounds showing that the optimal running time increases to $(c+1)^{tw}cdot n^{O(1)}$. We consider two variants depending on whether a copy of $H$ can be used multiple times in the packing. - If $H$ is not a clique, then the dependence of the running time on treewidth may not be even single exponential. Specifically, we show that if $H$ is any fixed graph where not every 2-connected component is a clique, then there is no $2^{o({tw}log {tw})}cdot n^{O(1)}$ algorithm for extsc{$H$-Partition}, assuming the Exponential-Time Hypothesis (ETH).

Problem

• Generalizing graph packing problems parameterized by treewidth
• Studying H-Packing and H-Partition on bounded-treewidth graphs
• Providing upper and lower bounds for clique and non-clique cases

Innovation

• Generalized packing with vertex reuse capacity constraints
• Treewidth parameterization for clique packing algorithms
• ETH-based lower bounds for non-clique components