Lower bounds for the integrality gap of the bi-directed cut formulation of the Steiner Tree Problem

TL;DR

This work introduces a Complete Metric (CM) formulation for the metric Steiner Tree Problem (STP) to address weaknesses of the Bi-directed Cut (BCR) model in metric instances. It proves several structural properties of the CM polytope, demonstrates a tight relationship between CM and BCR vertices, and shows how metric closure preserves STP values, enabling CM-based lower bounds for BCR gaps. The authors extend the Gap problem to STP and design two vertex-enumeration heuristics—One-Two-Costs (OTC) and Pure Half-Integer (PHI)—to uncover small, nontrivial instances with large integrality gaps (up to 19/18 for n≥10), providing new lower bounds for CM and BCR gaps on graphs with up to 10 nodes. They also discuss extensions to Pure One-Quarter (POQ) vertices and conjecture about exhaustiveness and structural properties of spanning PHI vertices. Collectively, the paper offers methodological advancements and computational benchmarks that inform tighter integrality-gap analyses and potential improvements in Steiner Tree approximations for metric graphs.

Abstract

In this work, we study the metric Steiner Tree problem on graphs focusing on computing lower bounds for the integrality gap of the bi-directed cut (BCR) formulation and introducing a novel formulation, the Complete Metric (CM) model, specifically designed to address the weakness of the BCR formulation on metric instances. A key contribution of our work is extending the Gap problem, previously explored in the context of the Traveling Salesman problems, to the metric Steiner Tree problem. To tackle the Gap problem for Steiner Tree instances, we first establish several structural properties of the CM formulation. We then classify the isomorphism classes of the vertices within the CM polytope, revealing a correspondence between the vertices of the BCR and CM polytopes. Computationally, we exploit these structural properties to design two complementary heuristics for finding nontrivial small metric Steiner instances with a large integrality gap. We present several vertices for graphs with a number of nodes <=10, which realize the best-known lower bounds on the integrality gap for the CM and the BCR formulations. We conclude the paper by presenting two new conjectures on the integrality gap of the BCR and CM formulations for small graphs.

Figures (4)

• Figure 1: Small STP instance with a large integrality gap: Skutella's graph with $n=15$, $t=8$, and $\alpha=\frac{8}{7}$konemann2011partitionbased. The hollow circle represents the root, the circles represent the terminals, and the squares represent the Steiner nodes. Every arc correspond to a variable $x_{ij}$ of value equal to $\frac{1}{4}$.
• Figure 2: Fractional vertices of $(7, 4)$, all with integrality gap $10/9$. Hollow circle: root; Circles: Terminals; Square: Steiner node. Note that the second, third, and fourth vertex belong to the same class of isomorphism, while the first one belongs to another class.
• Figure 3: Fractional vertices of $(8, 5)$, all with integrality gap 10/9. Hollow circle: root. Circles: Terminals. Squares: Steiner nodes.
• Figure 4: Fractional vertices of different gaps for different values of $(n,t)$. The first three vertices attain the maximum gap for their respective value of $(n,t)$.

Theorems & Definitions (28)

• Example 1
• Lemma 1
• Lemma 2
• proof
• Definition 1: Support graph
• Lemma 3
• proof
• Lemma 4
• proof
• Definition 2: Metric closure of a graph