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Optimal Routing across Constant Function Market Makers with Gas Fees

Author

Listed:
  • Carlos Escudero
  • Felipe Lara
  • Miguel Sama

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.

Suggested Citation

  • Carlos Escudero & Felipe Lara & Miguel Sama, 2026. "Optimal Routing across Constant Function Market Makers with Gas Fees," Papers 2603.02844, arXiv.org.
    • Handle: RePEc:arx:papers:2603.02844
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    References listed on IDEAS

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    1. Guillermo Angeris & Akshay Agrawal & Alex Evans & Tarun Chitra & Stephen Boyd, 2022. "Constant Function Market Makers: Multi-asset Trades via Convex Optimization," Springer Optimization and Its Applications, in: Duc A. Tran & My T. Thai & Bhaskar Krishnamachari (ed.), Handbook on Blockchain, pages 415-444, Springer.
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    4. Guillermo Angeris & Tarun Chitra & Alex Evans & Stephen Boyd, 2022. "Optimal Routing for Constant Function Market Makers," Papers 2204.05238, arXiv.org.
    5. Alberto Cambini & Laura Martein, 2009. "Generalized Convexity and Optimization," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-540-70876-6, March.
    6. Agostino Capponi & Ruizhe Jia, 2021. "The Adoption of Blockchain-based Decentralized Exchanges," Papers 2103.08842, arXiv.org, revised Jul 2021.
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