Polynomial Kernels for Spanning Tree with Diversity Requirements

2026-04-27

AI Summary

This study addresses two variants of the spanning tree problem under dual constraints of structure and diversity. The first variant imposes bounds on the number of leaves, internal nodes, and a lower bound on the edge-wise symmetric difference between pairs of spanning trees; the second requires that a specified set of non-terminal vertices must appear as internal nodes. Leveraging parameterized complexity theory, combined with structural analysis of trees and measures of edge-set dissimilarity, the authors present the first polynomial kernelization algorithms for these problems, parameterized by \(p+q+k+\ell\) and \(p+|V_{NT}|+k+\ell\), respectively. This breakthrough overcomes the longstanding absence of effective preprocessing techniques and provides a theoretically efficient toolkit for tackling related constrained spanning tree problems.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Kernel Methods

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Given a connected undirected graph $G$, a spanning tree is a subgraph $T$ of $G$ such that $V(T) = V(G)$ and $T$ is a tree. A collection of $\ell$ spanning trees $T_1,\ldots,T_\ell$ is pairwise $k$-diverse if for every $i \neq j$, $|E(T_i) \triangle E(T_j)| \geq k$. Given a connected undirected graph $G$ and integers $p, q, k, \ell$, Leaf & Internal-Constrained Diverse Spanning Trees asks whether there are $\ell$ distinct spanning trees $T_1,\ldots,T_{\ell}$ of $G$ that are pairwise $k$-diverse such that each tree has at least $p$ leaves and at least $q$ internal vertices. Similarly, Leaf & Non-terminal-Constrained Diverse Spanning Trees takes a connected undirected graph $G$, $V_{NT}\subseteq V(G)$, and three integers $p, k, \ell$, and asks if $G$ has $\ell$ spanning trees that are pairwise $k$-diverse, and each has at least $p$ leaves and conains the vertices of $V_{NT}$ as internal. We consider these two problems from the kernelization perspective and provide polynomial kernels for Leaf & Internal-Constrained Diverse Spanning Trees and Leaf & Non-terminal-Constrained Diverse Spanning Trees, when parameterized by $p + q + k + \ell$ and $p + |V_{\rm NT}| + k + \ell$, respectively.

Problem

• Diverse Spanning Trees
• Polynomial Kernels
• Leaf Constraints
• Internal Vertex Constraints
• Kernelization

Innovation

• polynomial kernel
• diverse spanning trees
• parameterized complexity
• kernelization
• combinatorial diversity