On the integrality gap of the Complete Metric Steiner Tree Problem via a novel formulation

2024-05-22

arXiv.org

AI Summary

This paper investigates the integrality gap lower bound of the Directed Cut (DCUT) formulation for the Metric Steiner Tree Problem. To overcome the limitation of DCUT—its difficulty in constructing high-gap instances on small graphs—we introduce a novel Completely Metric (CM) formulation and establish a structural correspondence between its polyhedral vertices and those of DCUT. We systematically classify CM vertex isomorphism classes and design two complementary heuristic algorithms to efficiently generate small-graph instances with large integrality gaps. Combining combinatorial optimization, polyhedral analysis, graph-theoretic modeling, and large-scale computational experiments, we obtain the current best-known integrality gap lower bounds for graphs with up to 10 vertices. This work marks the first gap analysis for Steiner trees under the CM model, revealing its theoretical advantages over DCUT. Finally, we propose three conjectures concerning integrality gaps of small graphs.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Security and Privacy: Large-scale security measurements

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 (DCUT) formulation and introducing a novel formulation, the Complete Metric (CM) model, specifically designed to address the weakness of the DCUT formulation on metric instances. A key contribution of our work is extending of 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 DCUT 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 $leq 10$, which realize the best-known lower bounds on the integrality gap for the CM and the DCUT formulations. We conclude the paper by presenting three new conjectures on the integrality gap of the DCUT and CM formulations for small graphs.

Problem

• Studying lower bounds for the Steiner Tree integrality gap
• Introducing a novel Complete Metric formulation model
• Designing heuristics to find high-gap Steiner instances

Innovation

• Novel Complete Metric formulation for Steiner Tree
• Structural analysis of polytope isomorphism classes
• Heuristics for finding high integrality gap instances