A.) First several pages are random fluff: Explains P, NP, NP-Hard, NP-Complete, history of P=NP, examples of NP-complete problems, examples of market efficiency, and other very uninteresting stuff that feels like it's trying to fill out pages.
B.) Makes this central argument:
1.) If markets are efficient, then no one can come up with a consistently profitable trading strategy based on analyzing historical data.
2.) Model historical data as a sequence of days where the market can go up or down on any given day.
3.) Model a trading strategy as being long, short, or neutral the market on any given day, to be started after recognizing some pattern for the previous N days.
4.) To determine from historical data whether a profitable strategy exists, you'd need to try all combinations of being long, short, or neutral over some time horizon, while looking for all market patterns. Trying all such combinations is O(3^N) or O(2^N), depending on whether you believe the expensive part is finding the strategy or finding the pattern.
5.) To verify that you have a winning strategy, you merely need to simulate the strategy over your historical data, which only takes O(N).
6.) Therefore, it is in NP. Similarly, it is NP-Hard because we can reduce 3-SAT to a market / trading problem. Therefore, "does a profitable strategy based on historical data exist?" is in NP-complete.
7.) Because our model shows finding the existence of a profitable strategy is NP-complete, then either P=NP or markets are inefficient.
C.) A few comments of my own:
1.) Suppose we believe the argument. The amount of historical data is finite, so maybe some hedge funds brute-forced a solution and removed the market inefficiencies they've found.
2.) I think SAT and travelling salesman are the wrong analogies, because solving both optimally can involve lots of backtracking. A* search on a landscape with reliable heuristics might be a better analogy. In trading markets, you don't need to try all 3^N combinations because just as in search algorithms, you have heuristics that can help prune large parts of your search space (if a strategy starts off losing money, no need to keep exploring all the 3^N combinations under it).
3.) The author assumes everyone represents and models the market his way. Maybe a more clever representation makes the problem tractable. Checkers is a solved game now, even though there are in theory infinitely many possible game histories to test.