Arbitrage algorithm for multiple exchanges, multiple currencies, and multiple amounts

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I'm searching for a way to apply an arbitrage algorithm across multiple exchanges, multiple currencies. and multiple trading amounts. I've seen examples of using BellmanFord and FloydWarshall, but the one's I've tried all seem to be assuming the graph data set is made up of prices for multiple currencies on one single exchange. I've tried tinkering and making it support prices across multiple exchanges but I haven't found any success.

One article I read said that I use BellmanFord and simply put only the best exchange's price in the graph (as opposed to all the exchange's prices). While it sounds like that should work, I feel like that could be missing out on value that way. Is this the right way to go about it?

And regarding multiple amounts, should I just make one graph per trade amount? So say I want to run the algorithm for $100 and for $1000, do I just literally populate the graph twice once for each set of data? The prices will be different at $100 than for $1000 so one exchange that has the best price at $100 may be different then that of the $1000 amount.

Examples:

The graph would look like this:

rates = [
    [1, 0.23, 0.26, 17.41],
    [4.31, 1, 1.14, 75.01],
    [3.79, 0.88, 1, 65.93],
    [0.057, 0.013, 0.015, 1],
]

currencies = ('PLN', 'EUR', 'USD', 'RUB')

REFERENCES:

algorithmic-trading path-finding graph-algorithm algorithm python
1 Answers

Trying for accuracy over speed, there's a way to represent the whole order book of each exchange inside of a linear program. For each bid, we have a real-valued variable that represents how much we want to sell at that price, ranging between zero and the amount bid for. For each ask, we have a real-valued variable that represents how much we want to buy at that price, ranging between zero and the amount asked for. (I'm assuming that it's possible to get partial fills, but if not, you could switch to integer programming.) The constraints say, for each currency aside from dollars (or whatever you want more of), the total amount bought equals the total amount sold. You can strengthen this by requiring detailed balance for each (currency, exchange) pair, but then you might leave some opportunities on the table. Beware counterparty risk and slippage.

For different amounts of starting capital, you can split dollars into "in-dollars" and "out-dollars" and constrain your supply of "in-dollars", maximizing "out-dollars", with a one-to-one conversion with no limit from in- to out-dollars. Then you can solve for one in-dollars constraint, adjust the constraint, and use dual simplex to re-solve the LP faster than from scratch.

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