Directionality and sign handling in multi-hop AMM routing

Determine, for multi-hop routing across automated market makers with intermediate tokens between an input token X and an output token Y, how to select the correct trade direction between each intermediate token pair and whether allocation amounts may change sign during the routing algorithm’s execution, thereby enabling a robust multi-hop extension of the concave-continuation-based transfer algorithm.

Background

The paper introduces concave continuation to extend AMM trade functions to negative inputs, unifying routing and arbitrage, and presents an extended transfer algorithm that solves the one-hop routing/arbitrage problem with convergence guarantees. This framework relies on local conservation laws and preserves concavity, allowing allocations to be negative to capture arbitrage.

While the one-hop setting is addressed, extending the approach to multi-hop routing (with intermediate tokens between the source and destination) introduces new challenges. Specifically, the appropriate trade direction between intermediate token pairs is not known a priori, and allocations may need to change sign during the algorithm’s progress. These issues must be resolved to construct a multi-hop transfer algorithm building on concave continuation.

References

Theoretically, concave continuation is the first step toward a multi-hop transfer algorithm: If there are some intermediate tokens M and N in the trade from token X to Y, we do not know a priori what the trade direction between M and N should be nor are we certain the allocation will flip signs or not during the routing algorithm's run. This remains future work.

— Concave Continuation: Linking Routing to Arbitrage  (2604.02909 - Jiang et al., 3 Apr 2026) in Discussion (Section 8)