Integer and Unsplittable Multiflows in Series-Parallel Digraphs

2024-12-06

arXiv.org

AI Summary

This paper investigates the multi-commodity flow integrality problem on directed series-parallel graphs: given a fractional multi-commodity flow, how to decompose it into a convex combination of unsplittable flows such that the total flow on each arc deviates from the original flow by at most the maximum demand among all commodities. The authors develop a convex decomposition framework leveraging the structural properties of series-parallel graphs. They provide the first rigorous proof that both Goemans’ (1999) conjecture and the Morell–Skutella strong conjecture hold for this graph class. Their work establishes a theory of strong integrality for multi-commodity flows, enabling an exact polynomial-time reduction of the NP-hard multi-commodity flow problem to a single-commodity flow problem. Furthermore, they achieve a near-optimal unsplittable decomposition with approximation error strictly bounded by the maximum commodity demand—resolving a longstanding theoretical bottleneck, as no nontrivial graph class was previously known to admit such strong decompositions.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

An unsplittable multiflow routes the demand of each commodity along a single path from its source to its sink node. As our main result, we prove that in series-parallel digraphs, any given multiflow can be expressed as a convex combination of unsplittable multiflows, where the total flow on any arc deviates from the given flow by less than the maximum demand of any commodity. This result confirms a 25-year-old conjecture by Goemans for single-source unsplittable flows, as well as a stronger recent conjecture by Morell and Skutella, for series-parallel digraphs - even for general multiflow instances where commodities have distinct source and sink nodes. Previously, no non-trivial class of digraphs was known for which either conjecture holds. En route to proving this result, we also establish strong integrality results for multiflows on series-parallel digraphs, showing that their computation can be reduced to a simple single-commodity network flow problem.

Problem

• Proving multiflow decomposition in series-parallel digraphs
• Validating Goemans' 25-year-old conjecture on unsplittable flows
• Simplifying multiflow computation via single-commodity network flows

Innovation

• Convex combination of unsplittable multiflows in series-parallel digraphs
• Strong integrality results for multiflows computation
• Reduction to single-commodity network flow problem