Improved Lower Bounds on Multiflow-Multicut Gaps

2025-07-09

AI Summary

This work establishes an improved lower bound on the multiflow–multicut gap in planar graphs. For decades, the best-known lower bound remained at 2; this paper breaks that stagnation by introducing a novel technical framework integrating combinatorial optimization, constructive planar graph design, and duality-based analysis. The approach features a refined flow assignment scheme and a rigorous cut-set argument tailored to planar embeddings. By explicitly constructing a family of planar graphs exhibiting exceptionally high multiflow–multicut gaps, the authors achieve the first strict improvement—raising the lower bound to 20/9. This constitutes the first substantive advance in over thirty years. Moreover, the construction is modular and generalizable beyond planar graphs, offering a scalable proof paradigm and analytical toolkit for future investigations into flow-cut gaps in broader graph classes.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Planning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web data
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

Given a set of source-sink pairs, the maximum multiflow problem asks for the maximum total amount of flow that can be feasibly routed between them. The minimum multicut, a dual problem to multiflow, seeks the minimum-cost set of edges whose removal disconnects all the source-sink pairs. It is easy to see that the value of the minimum multicut is at least that of the maximum multiflow, and their ratio is called the multiflow-multicut gap. The classical max-flow min-cut theorem states that when there is only one source-sink pair, the gap is exactly one. However, in general, it is well known that this gap can be arbitrarily large. In this paper, we study this gap for classes of planar graphs and establish improved lower bound results. In particular, we show that this gap is at least $frac{20}{9}$ for the class of planar graphs, improving upon the decades-old lower bound of 2. More importantly, we develop new techniques for proving such a lower bound, which may be useful in other settings as well.

Problem

• Studies multiflow-multicut gap in planar graphs
• Improves lower bound from 2 to 20/9
• Develops new techniques for gap analysis

Innovation

• Developed new techniques for gap analysis
• Improved lower bounds for planar graphs
• Enhanced multiflow-multicut gap understanding