# Maximum value of arbitrage

I'm trying to solve this from The Algorithm Design Manual:

6-23. Arbitrage is the use of discrepancies in currency-exchange rates to make a profit. For example, there may be a small window of time during which 1 U.S. dollar buys 0.75 British pounds, 1 British pound buys 2 Australian dollars, and 1 Australian dollar buys 0.70 U.S. dollars. At such a time, a smart trader can trade one U.S. dollar and end up with 0.75×2×0.7=1.05 U.S. dollars---a profit of 5%. Suppose that there are n currencies c1,...,cn and an n×n table R of exchange rates, such that one unit of currency ci buys R[i,j] units of currency cj. Devise and analyze an algorithm to determine the maximum value of R[c1,ci1]⋅R[ci1,ci2]⋯R[cik−1,cik]⋅R[cik,c1] Hint: think all-pairs shortest path.

My understanding is this is asking to find the longest simple cycle (LSC) on a directed weighted (complete?) graph. There is a reduction from longest simple path to LSC, so this would be NP-hard. It seems strange that the problem gives a hint, so maybe it's asking something else?.

The question is at least poorly worded, since it does not specify $$k \leq n$$ or require a simple cycle. So any positive-value arbitrage cycle can be "pumped" to generate a cycle with unbounded positive value.