Completely Independent Steiner Trees

2026-04-21

AI Summary

This study presents the first systematic investigation of the Completely Independent Steiner Tree (CIST) problem, which seeks to construct multiple Steiner trees for fault-tolerant communication such that the internal vertices and edges of the paths between terminals are mutually disjoint across trees. The authors unify the notions of completely independent spanning trees and internally disjoint Steiner trees, introduce a directed variant, and establish their equivalence. Through graph-theoretic analysis, structural characterizations, and complexity-theoretic arguments—complemented by dynamic programming and specialized algorithms for planar and bounded-treewidth graphs—the work proves the NP-hardness of the problem in general graphs, establishes a lower bound on graph connectivity sufficient for CIST existence, and designs polynomial-time algorithms for several restricted graph classes.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Computational social science
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Spanning trees are fundamental for efficient communication in networks. For fault-tolerant communication, it is desirable to have multiple spanning trees to ensure resilience against failures of nodes and edges. To this end, various notions of disjoint or independent spanning trees have been studied, including edge-disjoint, node/edge-independent, and completely independent spanning trees. Alongside these, several Steiner variants have also been investigated, where the trees are required to span a designated subset of vertices called terminals. For instance, the study of edge-disjoint spanning trees has been extended to edge-disjoint Steiner trees; a stronger variant is the problem of internally disjoint Steiner trees, where any two Steiner trees intersect exactly in the terminals. In this paper, we investigate the Steiner analogue of completely independent spanning trees, which we call \emph{completely independent Steiner trees}. A set of Steiner trees is completely independent if, for every pair of terminals $u,v$, the $(u,v)$-paths in all the Steiner trees are internally vertex-disjoint and edge-disjoint. This notion generalizes both completely independent spanning trees and internally disjoint Steiner trees. We provide a systematic study of completely independent Steiner trees from structural, algorithmic, and complexity-theoretic perspectives. In particular, we present several characterisations, connectivity bounds, algorithms, hardness results, and applications to special graph classes such as planar graphs and graphs of bounded treewidth. Along the way, we also introduce a directed variant of completely independent spanning trees via an equivalence with completely independent Steiner trees.

Problem

• completely independent Steiner trees
• Steiner trees
• internally vertex-disjoint
• edge-disjoint
• terminals

Innovation

• completely independent Steiner trees
• internally vertex-disjoint paths
• edge-disjoint paths
• graph connectivity
• parameterized algorithms