So an option on y=w1*S1+w2*S2+ .... is a functional of values
of y at current and later times but is it a functional of values
of any other stocks?
Consider stepping back on step in time to T-1 ?
we can exercise or not. If we exercise we get a payoff dependent
on y(T-1) and if we keep a payoff dependent on y(T). So where
does value of other stocks come in. Only if distribution of y(T)
as calculated from y(T-1) depends on these stocks. However
in simplest cases distribution of y(T) is independent of other
stock values at T-1. What is dependent is values of Y(T), s1(T) etc but
this seems to me irrelevant.
We can now go back recursively and deduce:
that an American option in y can be calculated one dimensionally.

We can do an estimate of computational complexity of 100 stocks
of this type
Suppose in very round numbers
50 times values
100 stocks
10,000 independent paths
generate path is 50 floating point calcs per stock per time (uncertain)

So if correlated assume diagonalize correlation matrix before one starts
Further assume simple correlated Gaussians with constant parameters
so we use simple program and not current general Metropolis in production
(We can test ideas with current program)
Then diagonalizing correlation matrix and finding a single variable
American Option take trivial time
Actual path generation is 2.5 giga floating point operations
This takes 50 seconds on a 50 megaflop machine (PC?) which is
less than target 5 minutes
Note there is lots of parallelism (each stock can be calculated
independently and there is parallelism over paths for each stock with
this simple non Metropolis approach)