Structural Parameterization of Steiner Tree Packing

2025-05-14

AI Summary

Steiner Tree Packing (STP) is a fundamental NP-hard problem in VLSI design. This paper establishes, for the first time, fixed-parameter tractability (FPT) of STP with respect to structural graph parameters: specifically, it proves FPTness parameterized by both tree-cut width and fracture number, and significantly improves the time complexity under the latter parameter. Furthermore, it generalizes Edge-Disjoint Paths (EDP) techniques to Generalized Steiner Tree Packing (GSTP), resolving the long-standing open question of whether EDP on augmented graphs is FPT when parameterized by tree-cut width. Technically, the approach integrates augmented graph construction, slim tree-cut width analysis, fracture number theory, dynamic programming, and modular decomposition. The results provide the first rigorous structural complexity characterization of STP grounded in graph-theoretic parameters and unify and extend prior FPT results for EDP to the broader GSTP framework.

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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Steiner Tree Packing (STP) is a notoriously hard problem in classical complexity theory, which is of practical relevance to VLSI circuit design. Previous research has approached this problem by providing heuristic or approximate algorithms. In this paper, we show the first FPT algorithms for STP parameterized by structural parameters of the input graph. In particular, we show that STP is fixed-parameter tractable by the tree-cut width as well as the fracture number of the input graph. To achieve our results, we generalize techniques from Edge-Disjoint Paths (EDP) to Generalized Steiner Tree Packing (GSTP), which generalizes both STP and EDP. First, we derive the notion of the augmented graph for GSTP analogous to EDP. We then show that GSTP is FPT by (1) the tree-cut width of the augmented graph, (2) the fracture number of the augmented graph, (3) the slim tree-cut width of the input graph. The latter two results were previously known for EDP; our results generalize these to GSTP and improve the running time for the parameter fracture number. On the other hand, it was open whether EDP is FPT parameterized by the tree-cut width of the augmented graph, despite extensive research on the structural complexity of the problem. We settle this question affirmatively.

Problem

• Develops FPT algorithms for Steiner Tree Packing using structural graph parameters
• Generalizes Edge-Disjoint Paths techniques to Generalized Steiner Tree Packing
• Solves open question on FPT for tree-cut width in augmented graphs

Innovation

• FPT algorithms for STP via structural parameters
• Generalized techniques from EDP to GSTP
• Augmented graph concept for GSTP analysis