Optimal Routing across Constant Function Market Makers with Gas Fees

TL;DR

This paper develops a mathematical framework for optimal trade routing in decentralized exchanges with constant function market makers, accounting for fixed gas fees and heterogeneous pools, extending existing models to more realistic on-chain costs.

Contribution

It introduces a mixed-integer optimization model with necessary and sufficient optimality conditions, accommodating non-convex invariant functions and fixed gas fees in routing problems.

Findings

Derived explicit optimality conditions linking prices, fees, and activation thresholds.

Established bounds on utility loss due to model relaxation.

Extended mathematical theory for routing in fragmented markets with gas fees.

Abstract

We study the optimal routing problem in decentralized exchanges built on Constant Function Market Makers when trades can be split across multiple heterogeneous pools and execution incurs fixed on-chain costs (gas fees). While prior routing formulations typically abstract from fixed activation costs, real on-chain execution presents non-negligible gas fees. They also become convex under concavity/convexity assumptions on the invariant functions. We propose a general optimization framework that allows differentiable invariant functions beyond global convexity and incorporates fixed gas fees through a mixed-integer model that induces activation thresholds. Subsequently, we introduce a relaxed formulation of this model, whereby we deduce necessary optimality conditions, obtaining an explicit Karush-Kuhn-Tucker system that links prices, fees, and activation. We further establish sufficient optimality conditions using tools from generalized convexity (pseudoconcavity/pseudoconvexity and quasilinearity), yielding a verifiable optimality characterization without requiring convex trade functions. Finally, we relate the relaxed solution to the original mixed-integer model by providing explicit approximation bounds that quantify the utility gap induced by relaxation. Our results extend the mathematical theory for routing by offering no-trade conditions in fragmented on-chain markets in the presence of gas fees.