Structural Parameterization of Steiner Tree Packing

TL;DR

This paper introduces the first fixed-parameter tractable algorithms for Steiner Tree Packing based on structural graph parameters, extending techniques from Edge-Disjoint Paths and improving understanding of problem complexity.

Contribution

It presents novel FPT algorithms for STP parameterized by tree-cut width and fracture number, generalizing methods from EDP and resolving open questions about EDP's complexity.

Findings

STP is FPT by tree-cut width and fracture number.

GSTP can be solved efficiently using augmented graphs.

The paper resolves the open problem of EDP's FPT status by tree-cut width.

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.