An Efficient Algorithm for Optimal Routing Through Constant Function Market Makers

Abstract

Constant function market makers (CFMMs) such as Uniswap have facilitated trillions of dollars of digital asset trades and have billions of dollars of liquidity. One natural question is how to optimally route trades across a network of CFMMs in order to ensure the largest possible utility (as specified by a user). We present an efficient algorithm, based on a decomposition method, to solve the problem of optimally executing an order across a network of decentralized exchanges. The decomposition method, as a side effect, makes it simple to incorporate more complicated CFMMs, or even include ‘aggregate CFMMs’ (such as Uniswap v3), into the routing problem. Numerical results show significant performance improvements of this method, tested on realistic networks of CFMMs, when compared against an off-the-shelf commercial solver.

A Closed Form Solutions

Here, we cover some of the special cases where it is possible to analytically write down the solutions to the arbitrage problems presented previously.

Geometric Mean Trading Function. Some of the most popular swap markets, for example, Uniswap v2 and most Balancer pools, which total over $2B in reserves, are geometric mean markets (5) with $n=2$. This trading function can be written as

$$ \varphi (R) = R_1^wR_2^{1-w}, $$

where $0 < w < 1$ is a fixed parameter. This very common trading function admits a closed-form solution to the arbitrage problem (8). Using (12), we can write

$$ f_1(\delta _1) = R_2\left( 1- \left( \frac{1}{1 + \gamma \delta _1/R_1}\right) ^\eta \right) $$

where $\eta = w/(1-w)$. (A similar equation holds for $f_2$.) Using (15) and (16), and defining

$$ \delta _1 = \frac{R_1}{\gamma }\left( \left( \eta \gamma \frac{\nu _2}{\nu _1}\frac{R_2}{R_1}\right) ^{1/(\eta +1)} - 1\right) , $$

we have that $\delta _1^\star = \max \{\delta _1, 0\}$ is an optimal point for (14). Note that when we take $w = 1/2$ then $\eta = 1$ and we recover the optimal arbitrage for Uniswap given in [6, App. A].

Bounded Liquidity Variation. The bounded liquidity variation (4) of the product trading function satisfies the definition of bounded liquidity given in §4.1, whenever $\alpha , \beta > 0$. We can write the forward exchange function for the bounded liquidity product function (4), using (12), as

$$ f_1(\delta ) = \min \left\{ R_2, \frac{\gamma \delta (R_2 + \beta )}{R_1 + \gamma \delta + \alpha }\right\} $$

The ‘min’ here comes from the definition of a CFMM: it will not accept trades which pay out more than the available reserves. The maximum amount that a user can trade with this market, which we will write as $\delta _1^-$, is when $f_1(\delta _1^-) = R_2$, i.e.,

$$ \delta _1^- = \frac{1}{\gamma }\frac{R_2}{\beta }(R_1 + \alpha ). $$

(Note that this can also be derived by taking $f_1(\delta _1) = R_2$ in (12) with the invariant (4).) This means that

$$ f_1^-(\delta _1^-) = \gamma \frac{\beta ^2}{(R_1 + \alpha )(R_2 + \beta )}, $$

is the minimum supported price for asset 1. As before, a similar derivation yields the case for asset 2. Writing $k=(R_1+ \alpha )(R_2 + \beta )$, we see that we only need to solve (14) if the price $\nu _1/\nu _2$ is in the active interval (17),

$$\begin{aligned} \frac{\gamma \beta ^2}{k} < \frac{\nu _1}{\nu _2} < \frac{k}{\gamma \alpha ^2}. \end{aligned}$$

(18)

Otherwise, we know one of the two ‘boundary’ solutions, $\delta _1^-$ or $\delta _2^-$, suffices.