Approximate counting and NP search problems

TL;DR

This paper introduces a new class of NP search problems based on approximate counting, characterizes it computationally, and explores its limitations within bounded arithmetic, advancing understanding of logical hierarchies.

Contribution

It defines a new class of NP search problems provable by combinatorial approximate counting and analyzes its boundaries within bounded arithmetic using novel technical tools.

Findings

The class includes Ramsey and weak pigeonhole search problems.

Relative to an oracle, it does not contain the CPLS problem.

It shows that certain bounded arithmetic provability does not extend to all -complexity sentences.

Abstract

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 $\mathrm{APC}_2$ of [Je\v{r}\'abek 2009]. 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 $T^2_2$, this shows that $\mathrm{APC}_2$ does not prove every $\forall \Sigma^b_1$ sentence which is provable in bounded arithmetic. This answers the question posed in [Buss, Ko{\l}odziejczyk, Thapen 2014] and represents some progress in the programme 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 [Pudl\'ak, Thapen 2017], 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 $\textrm{P}^{\textrm{NP}}$ 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.