The Formal Mathematical Falsification of the Dinitz-Garg-Goemans Cost Conjecture

 

The Dinitz-Garg-Goemans cost conjecture (1999) for the Single-Source Unsplittable Flow Problem (SSUFP) postulated that converting an optimal fractional flow into an unsplittable flow within standard capacity bounds is always achievable without increasing total costs.

 

This paper presents a formal, purely algebraic falsification of this conjecture in static space. Rather than relying on complex, numerical black-box constructs, we introduce a minimal, precisely parameterized family of asymmetric graph topologies. This approach ensures that the structural failure of the conjecture remains immediately transparent, reproducible, and formally verifiable for the mathematical community during peer review. We analytically deduce that the unsplittable cost coefficient \(\rho \) diverges to infinity under escalating boundary costs (\(\lim_{H \to \infty} \rho = \infty\)).

 

 

 

Building upon this rigorous foundation, the second part proposes a constructive system expansion. By introducing an infinitesimal temporal component — a singular storage arc at the source — the unsplittable demands are orthogonalized across a discrete time horizon.

 

This interdisciplinary approach completely smoothes the topological singularity of the static domain. The dynamic cost coefficient is proven to converge toward a fixed, predictable upper bound (\(\lim_{H \to \infty} \rho_{\text{time}} = 1 + \frac{P}{2L} < \infty\)) without violating the formal flow restrictions. This work closes the theoretical gap of the static SSUFP and provides a mathematical blueprint for optimizing real-world, time-extended distribution networks.