Time/Space Optimal k-Subset Operator Application - Is this a named problem? I have searched extensively and unsuccessfully for references to a
combinatorial problem that arises in my work. I am hoping someone
can tell me if this type of problem has a "name" and provably optimal solutions.

The problem: Given a set of objects denoted by $X_1,\ldots,X_N$, a commutative
and associative operator $\oplus$, and $K \leq N$, generate all $N \choose K$
subset-applications of the operator in a way that first minimizes the number of
operator applications, and then minimizes the number of partial results that
must be stored, assuming that once a complete combination is generated its
storage can be recovered. For example, for $5 \choose 4$ we need to generate the five
new objects
$$
X_1 \oplus X_2 \oplus X_3 \oplus X_4, \\
X_1 \oplus X_2 \oplus X_3 \oplus X_5, \\
X_1 \oplus X_2 \oplus X_4 \oplus X_5, \\
X_1 \oplus X_3 \oplus X_4 \oplus X_5, \\
X_2 \oplus X_3 \oplus X_4 \oplus X_5 \\
$$
which naively requires 15 operator applications and a single
accumulator. However a better solution
requires only 11 operator applications if storage is allocated for 3 partial
results.

Note that no negation or inverse of the operator $\oplus$ is assumed.  This rules
out using most popular combination-generation schemes such as Gray codes. In
the motivating application the operator is also idempotent ($X \oplus X = X$)
but I don't think this fact is helpful to a solution.

I have developed a good algorithm to solve the problem
using at most $N$ accumulators.  However it strikes me that others have
probably analyzed this type of problem and I simply don't know the correct
terminology to guide my search.
For $K \ll 2N/3$, a good solution goes as follows, for $K$ even (the case $K$ odd is very similar):


Generate all XORs of length 2.
Generate all XORs of length 3.
...
Generate all XORs of length $K/2$.
Generate all XORs of length $K$.


This requires $\binom{N}{K/2} + \binom{N}{K/2-1} \approx \binom{N}{K/2}$ memory and $\binom{N}{2} + \cdots + \binom{N}{K/2} + \binom{N}{K} \approx \binom{N}{K}$ operations. This is close to optimal in terms of number of operations, since clearly at least $\binom{N}{K}$ are needed.

You can optimize this a bit as follows:


Generate all XORs of length 2 of $X_1,\ldots,X_{N-K/2+2}$.
Generate all XORs of length 3 of $X_1,\ldots,X_{N-K/2+3}$.
...
Generate all XORs of length $K/2$ of $X_1,\ldots,X_N$.
Generate all XORs of length $K$.


The savings are not significant.