Approximate counting and NP search problems

0565597 - MÚ 2023 RIV SG eng J - Journal Article
Kołodziejcz, L. A. - Thapen, Neil
Approximate counting and NP search problems.
Journal of Mathematical Logic. Roč. 22, č. 3 (2022), č. článku 2250012. ISSN 0219-0613. E-ISSN 1793-6691
R&D Projects: GA ČR(CZ) GA19-05497S
EU Projects: European Commission(XE) 339691 - FEALORA
Institutional support: RVO:67985840
Keywords : approximate counting * bounded arithmetic * CPLS * NP search problems
OECD category: Pure mathematics
Impact factor: 0.9, year: 2022
Method of publishing: Limited access
https://doi.org/10.1142/S021906132250012X

We study a new class of NP search problems, those which can be proved total using standard combinatorial reasoning based on approximate counting. Our model for this kind of reasoning is the bounded arithmetic theory APC2 of [E. Jeřábek, Approximate counting by hashing in bounded arithmetic, J. Symb. Log. 74(3) (2009) 829-860]. In particular, the Ramsey and weak pigeonhole search problems lie in the new class. We give a purely computational characterization of this class and show that, relative to an oracle, it does not contain the problem CPLS, a strengthening of PLS. As CPLS is provably total in the theory T22, this shows that APC2 does not prove every ς1b sentence which is provable in bounded arithmetic. This answers the question posed in [S. Buss, L. A. Kołodziejczyk and N. Thapen, Fragments of approximate counting, J. Symb. Log. 79(2) (2014) 496-525] and represents some progress in the program of separating the levels of the bounded arithmetic hierarchy by low-complexity sentences. Our main technical tool is an extension of the 'fixing lemma' from [P. Pudlák and N. Thapen, Random resolution refutations, Comput. Complexity, 28(2) (2019) 185-239], a form of switching lemma, which we use to show that a random partial oracle from a certain distribution will, with high probability, determine an entire computation of a PNP oracle machine. The introduction to the paper is intended to make the statements and context of the results accessible to someone unfamiliar with NP search problems or with bounded arithmetic.
Permanent Link: https://hdl.handle.net/11104/0337115