Structural Parameterization of Steiner Tree Packing

TL;DR

The paper studies STP, NP-hard in general, through structural parameterization by introducing GSTP to unify STP and EDP. It develops the augmentation framework to transform instances into augmented graphs and proves $FPT$ algorithms for GSTP parameterized by augmented fracture number and augmented/slim tree-cut width, with implications for STP and EDP. Key techniques include ILP encodings, component configurations, and dynamic programming on simple/friendly tree-cut decompositions, enabling sub-exponential dependence on the parameter and, in some cases, doubly-exponential bounds. The results advance exact, parameterized tractability for packing problems on graphs, offering practical pathways for circuit design and network multicast applications where graph structure can be exploited. Overall, the work settles several open questions about $FPT$-tractability of GSTP/EDP/STP under tree-cut and fracture-type parameters and suggests directions for further tightening runtimes and extending augmentation-based methods to related problems.

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.

Figures (4)

• Figure 1: An overview of common structural parameters and their relation. We draw an edge from a parameter $\alpha$ to a parameter $\beta$, if given $\alpha$ we can compute an upper bound on $\beta$. A Hardness result for a parameter gives a similar results for all ancestors; an $\mathop{\mathrm{\mathsf{FPT}}}\nolimits$-algorithm for a parameter gives an $\mathop{\mathrm{\mathsf{FPT}}}\nolimits$-algorithm for all descendants.
• Figure 3: The gadget ensuring that at most one edge adjacent to $m_i$ is used in the solution.
• Figure 4: A family of graphs with tree-cut width at most 5. In the depicted nice tree-cut decomposition the node $m$ has $\ell + 2$ bold children, where $\ell$ can be chosen freely.
• Figure 5: A family of host graphs with a terminal set increasing the tree-cut width of the augmented graph without bound.

Theorems & Definitions (73)

• Definition 1
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