Greedy Distributed Optimization of Multi-Commodity Flows Draft --- not for distribution

While multi-commodity flow is a classical combinatorial optimization problem, it also directly addresses a number of practically important issues of congestion and bandwidth management in connection-oriented network architectures. We consider solutions for distributed multi-commodity flow problems, which are solved by multiple agents operating in a cooperative but uncoordinated manner. We provide first known stateless distributed optimization algorithms for the concurrent multi-commodity flow problem with polylogarithmic convergence. More precisely, our algorithm achieves 1 + approximation, with running time O(log P · logm · (1/ )) (P is number of paths in the network). No prior results exist for our model. Viewed from the point of view of classical sequential algorithm, our algorithm is a reasonable alternative to existing polynomial-time sequential approximation algorithms, such as Garg-Könemann [GK98]. The algorithm is rather elegant, (approximately a dozen lines of pseudo-code), and can be be easily implemented or taught in a classroom. Our result can be contrasted with numerous (thousands) of distributed optimization heuristics for multi-commodity flow published in the course of last four decades, that lack non-trivial provable performance guarantees. Remarkably, our algorithm requires that the increase in the flow rate on a link is more aggressive than the decrease in the rate. Essentially all of existing flow control heuristics are variations of TCP, which uses a conservative cap on the increase (e.g., additive), and rather liberal cap on the decrease (e.g., multiplicative). In contrast, our algorithms requires the increase to be multiplicative, and that this increase is dramatically more aggressive than the decrease in the rate. We conclude that rigorous analysis does suggest a drastic change to existing networking building blocks. ∗Johns Hopkins University. email: [email protected]. Partially supported by NSF grants CCF 0515080, ANIR-0240551, and CCR-0311795, and CNS-0617883. †IBM T.J. Watson Research Center. email: [email protected].