Year: 2024
Type: article
Abstract: Recently, Pasarkar et al. (2023) have introduced the new $${{\,\mathrm{\texttt{TFNP}}\,}}$$ TFNP subclass called $${{\,\mathrm{\texttt{PLC}}\,}}$$ PLC that contains the class $${{\,\mathrm{\texttt{PPP}}\,}}$$ 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 $${{\,\mathrm{\texttt{PLC}}\,}}$$ PLC . This paper discusses the complexity of the three generalizations of Constrained Long Choice, a $${{\,\mathrm{\texttt{PLC}}\,}}$$ PLC -complete problem. We first discuss the parallel variant: Given a Long Choice instance and $$\varvec{m}$$ m beginning elements, find $$\varvec{m}$$ m Constrained Long Choice solutions. We show that this variant is $${{\,\mathrm{\texttt{FP}}\,}}_{\Vert }^{{{\,\mathrm{\texttt{PLC}}\,}}}$$ 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 $$\varvec{T}$$ T , find $$\varvec{T}$$ T Long Choice solutions satisfying the suitable conditions. We prove that this variant is $${{\,\mathrm{\texttt{FP}}\,}}^{{{\,\mathrm{\texttt{PLC}}\,}}}$$ FP PLC -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 $${{\,\mathrm{\texttt{PLC}}\,}}$$ PLC -hard but also $${{\,\mathrm{\texttt{PLS}}\,}}$$ PLS -hard, where the class $${{\,\mathrm{\texttt{PLS}}\,}}$$ 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.
Source: Theory of Computing Systems
Author: Takashi Ishizuka
Institution: Fujitsu (Japan)
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
Funder: ACT-X
Award: G8707452894
OpenAlex ID: W4405839378