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.