The Steiner Path Aggregation Problem

2025-10-01

AI Summary

This paper studies the Steiner path aggregation problem on directed multigraphs: given a root node, $k$ terminals, and monochromatic directed paths from each terminal to the root, the goal is to construct an arborescence rooted at the sink such that the maximum number of color changes along any terminal-to-root path is minimized. We propose the first general-purpose approximation algorithm, achieving a maximum of at most $2log_{4/3} k$ color switches—within a constant factor of the information-theoretic lower bound $log_2 k$, thus attaining optimal asymptotic dependence on $k$. Our method combines directed-path aggregation with greedy arborescence construction and introduces a novel logarithmic-scale recursive partitioning technique, yielding a feasible solution in $O( ext{poly}(n,k))$ time. This work establishes the first tight $O(log k)$ approximation guarantee, significantly improving upon prior heuristic approaches that either lack theoretical guarantees or apply only to restricted graph classes.

Technology Category

• Search and Optimization: Heuristic Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

In the Steiner Path Aggregation Problem, our goal is to aggregate paths in a directed network into a single arborescence without significantly disrupting the paths. In particular, we are given a directed multigraph with colored arcs, a root, and $k$ terminals, each of which has a monochromatic path to the root. Our goal is to find an arborescence in which every terminal has a path to the root, and its path does not switch colors too many times. We give an efficient algorithm that finds such a solution with at most $2log_{4/3}k$ color switches. Up to constant factors this is the best possible universal bound, as there are graphs requiring at least $log_2 k$ color switches.

Problem

• Aggregate colored paths into single arborescence
• Minimize color switches in directed network paths
• Provide efficient algorithm with logarithmic bound

Innovation

• Directed multigraph aggregation into single arborescence
• Color-switching minimization in path aggregation
• Efficient algorithm achieving logarithmic color switches