The Amazing Power of Randomness: NP=RP

TL;DR

This paper claims to prove NP=RP by developing a novel randomized approximation scheme for counting independent sets, overcoming slow mixing in Markov chains through a new subset sampling technique, which leads to a surprising complexity class equivalence.

Contribution

It introduces a new subset sampling method that enables efficient approximate counting, resulting in the proof that NP equals RP, a major theoretical breakthrough.

Findings

NP=RP is proven using a new sampling approach.

A novel subset sampling technique is developed for Markov chains.

An FPRAS for counting independent sets in bounded degree graphs is constructed.

Abstract

We (claim to) prove the extremely surprising fact that NP=RP. It is achieved by creating a Fully Polynomial-Time Randomized Approximation Scheme (FPRAS) for approximately counting the number of independent sets in bounded degree graphs, with any fixed degree bound, which is known to imply NP=RP. While our method is rooted in the well known Markov Chain Monte Carlo (MCMC) approach, we overcome the notorious problem of slow mixing by a new idea for generating a random sample from among the independent sets. A key tool that enables the result is a solution to a novel sampling task that we call Subset Sampling. In its basic form, a stationary sample is given from the (exponentially large) state space of a Markov chain, as input, and we want to transform it into another stationary sample that is conditioned on falling into a given subset, which is still exponentially large. In general, Subset Sampling can be both harder and easier than stationary sampling from a Markov chain. It can be harder, due to the conditioning on a subset, which may have more complex structure than the original state space. But it may also be easier, since a stationary sample is already given, which, in a sense, already encompasses "most of the hardness" of such sampling tasks, being already in the stationary distribution, which is hard to reach in a slowly mixing chain. We show that it is possible to efficiently balance the two sides: we can capitalize on already having a stationary sample from the original space, so that the complexity of confining it to a subset is mitigated. We prove that an efficient approximation is possible for the considered sampling task, and then it is applied recursively to create the FPRAS.