Research Update: A Geometric Approach for Optimal Channel Rebalancing

This investigation is grounded on the foundational work presented in earlier discussions and documented resources concerning channel depletion, topology cycles, and the mathematical theory underpinning payment channel networks. The primary motivation behind this study stems from the practical challenges associated with off-chain rebalancing, which, despite not affecting theoretical payment feasibility rates, has been shown to cause frequent routing failures due to unbalanced channels.

This research proposes a structured, globally optimal strategy for channel rebalancing, aimed at addressing these inefficiencies by starting from a network state characterized by severe liquidity depletion. By formulating the rebalancing task as a constrained optimization problem, the study employs both continuous relaxation methods through Quadratic Programming and Integer Linear Programming for ensuring adherence to satoshi constraints and preventing financial losses among nodes. This computational approach endeavors to find the nearest feasible liquidity state that aligns with a predefined target, such as achieving balanced channels across the network.

The findings from applying this methodology to a heavily depleted network are compelling. Prior to optimization, only 11.31% of channels exhibited balanced liquidity. Post-optimization, this percentage surged to 56.30%, with moderately imbalanced channels also seeing a significant increase in their proportion. The optimizer recommended reallocating 33.23% of the total network capacity to achieve these results, demonstrating the potential effectiveness of this rebalancing strategy in mitigating liquidity issues across the network.

Despite the promising outcomes, the study acknowledges several limitations and open questions that warrant further exploration. Among these are the central coordination requirement, the computational cost associated with solving optimization problems at scale, and the challenge of defining a universally desirable target state for channel balances. Additionally, the research opens up inquiries into agent-based approximations and the viability of continuing this line of thought, especially considering the current tendency among node operators to manage liquidity through fees.

In summary, the notebook not only showcases the application of advanced mathematical and computational techniques to optimize liquidity in payment channel networks but also sets the stage for ongoing dialogue and experimentation within the community. The code made available for public use invites further engagement, analysis, and potentially, the development of more practical solutions tailored to the dynamic needs of node operators and network participants.