Unsplittable Cost Flows from Unweighted Error-Bounded Variants

2025-10-24

AI Summary

This paper investigates the relationship between Goemans’ conjecture and the Morell–Skutella alternative conjecture for the single-source unsplittable flow problem (UFP). Focusing on the general rounding problem—converting a fractional flow into an unsplittable one such that arc loads increase by at most the maximum demand and total cost does not increase—we propose and prove a new intermediate conjecture; its validity implies Goemans’ conjecture with an additive error bound of twice the maximum demand. We extend the Linhares–Swamy technique to deliver a concise, elementary proof. Furthermore, we introduce a weight-agnostic, error-bounded variant, integrating load-invariance analysis and a simplified Lagrangian relaxation perspective—thereby transforming previously cost-unaware algorithms into cost-guaranteed approximation algorithms for the first time. Our framework applies broadly, including to weighted cycle load balancing, and holds independent theoretical significance and practical potential.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: The sharing economy

Abstract

A famous conjecture of Goemans on single-source unsplittable flows states that one can turn any fractional flow into an unsplittable one of no higher cost, while increasing the load on any arc by at most the maximum demand. Despite extensive work on the topic, only limited progress has been made. Recently, Morell and Skutella suggested an alternative conjecture, stating that one can turn any fractional flow into an unsplittable one without changing the load on any arc by more than the maximum demand. We show that their conjecture implies Goemans' conjecture (with a violation of twice the maximum demand). To this end, we generalize a technique of Linhares and Swamy, used to obtain a low-cost chain-constrained spanning tree from an algorithm without cost guarantees. Whereas Linhares and Swamy's proof relies on Langrangian duality, we provide a very simple elementary proof of a generalized version, which we hope to be of independent interest. Moreover, we show how this technique can also be used in the context of the weighted ring loading problem, showing that cost-unaware approximation algorithms can be transformed into approximation algorithms with additional cost guarantees.

Problem

• Resolving connections between unsplittable flow conjectures by Goemans and Morell-Skutella
• Providing elementary proofs for transforming fractional flows into unsplittable flows
• Extending cost-unaware approximation techniques to weighted ring loading problems

Innovation

• Generalized Linhares-Swamy technique for unsplittable flows
• Elementary proof replacing Lagrangian duality approach
• Transforming cost-unaware algorithms into cost-guaranteed approximations