Functional design of efficient and parallelizable combinatorial generators using convolution

The application of program transformation and algebraic methods to the development of efficient combinatorial optimization (CO) algorithms relies on an exhaustive combinatorial generator for the problem specification, followed by the fusion of thinning or filtering processes into this specification. However, the effectiveness of such fusion transformations critically depends on the structural compatibility between the objective function and the generator, which is highly problem dependent. In practice, when the majority of candidate solutions remain unfiltered or are not eliminated-as is the case for most intractable CO problems-the overall efficiency of the resulting fused program is largely determined by the intrinsic efficiency of the combinatorial generator. Consequently, if the specification itself exhibits suboptimal performance, the fused program will inherit a correspondingly inferior level of efficiency. We argue that a genuine designed process should also account for hardware compatibility and parallelizability-particularly the ability to support efficient parallel execution on modern hardware architectures, including multi-level cache hierarchies and GPUs. However, does achieving formal correctness necessarily conflict with designing algebraically elegant algorithms that support fusion? Can we obtain both simultaneously? In this paper, we show that techniques from functional programming, provide powerful formal tools for the systematic construction of such hardware-compatible and parallelizable combinatorial generators. This paper investigates generators for two of the most fundamental combinatorial structures-combinations and permutations-together with their natural extension to nested generators (e.g., combinations/permutations of combinations/permutations).