Recently, Pasarkar et al. (2023) have introduced the new TFNP subclass called PLC that contains the class PPP; they also have proven that several search problems related to extremal combinatorial principles (e.g., Ramsey’s theorem and the sunflower lemma) belong to PLC. This paper discusses the complexity of the three generalizations of Constrained Long Choice, a PLC-complete problem. We first discuss the parallel variant: Given a Long Choice instance and m beginning elements, find mConstrained Long Choice solutions. We show that this variant is FP‖PLC-complete. Next, we discuss the iterative variant: Given a Constrained Long Choice instance, a process function that specifies the next beginning element depending on the current solution, and an iteration parameter T, find TLong Choice solutions satisfying the suitable conditions. We prove that this variant is FPPLC-complete. Finally, we consider the inductive variant, which is an iterative variant with the inductive principle. We prove that the inductive variant is not only PLC-hard but also PLS-hard, where the class PLS is the set of all search problems that are solvable by a local search method. Furthermore, we show that our iterative and inductive variants are closed under the Turing reduction.
Read full abstract