Approximate counting and NP search problems
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We study a new class of NP search problems, those which can be proved total in the theory APC_2 of [Jeřábek 2009]. This is an axiomatic theory in bounded arithmetic which can formalize standard combinatorial arguments based on approximate counting. In particular, the Ramsey and weak pigeonhole search problems lie in the 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 APC_2 does not prove every ∀Σ^b_1 sentence which is provable in bounded arithmetic. This answers the question posed in [Buss, Kołodziejczyk, 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 [Pudlák, 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 P^NP oracle machine. The paper is intended to be accessible to someone unfamiliar with NP search problems or with bounded arithmetic.
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