# Questions tagged [pcp]

Probabilistically checkable proofs

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### Inapproximability hardness through reductions [closed]

I am trying to understand hardness of approximation and following the book "Approximation Algorithms" by Vijay V. Vazirani. This question is related to Section 29.1 of the book. The book ...
326 views

### Technical lemma about curves used in original proof of PCP theorem

I am reading the proof from here and found a technical lemma that seems to be incorrect (its proof is short and very vague). I know this is rather specific and the context is problematic, but I couldn'...
206 views

### Results comparing BQP and NEXP

Are there oracle results with $$P=NP\neq BQP=NEXP\mbox{ and }P=NP\neq BQP\neq NEXP?$$ Also is there a $PCP$ characterization of $BQP$ like $$PCP(O(poly(n)),1)=PCP(O(poly(n)),O(poly(n)))=NEXP?$$
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### Technical issue with PCP theorem proof

I am reading the proof from here and I stumbled upon a technical (yet crucial) problem. I know this is rather specific and the context is problematic, but I couldn't figure it out myself. In pages 51 ...
174 views

### Best known asymptotic PCP sizes / 3-SAT

What are the best known asymptotic upper bounds on sizes of probabilistically checkable proofs? Ideally, I am looking for a contemporary survey on this broad question, but if there is none, I am ...
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### Does $\textbf{PCP}[poly(n), O(1)] = \textbf{coRP}$?

Something has been buzzing me recently. It is well-known that $\textbf{PCP}[poly(n), 0] = \textbf{coRP}$, but does $\textbf{PCP}[poly(n), O(1)] = \textbf{coRP}$ ? I have found a proof for this ...
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### Proof of Majority is stablest in “reverse” in the MAXCUT hardness paper by Khot et al

This is about Proposition 7.4 here. I think there is a slight error in the proof of this proposition. Basically, authors have taken $g$ to be the odd part of the function $f$. Due to which we can say ...
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### PCP research proposal [closed]

Hi I am taking an undergraduate taking a course in Probabilistic checkable Proofs. I will greatly appreciate if you can suggest some good research ideas and pertaining reading for someone who is just ...
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### Non-trivial PCP characterizations of complexity classes beyond ELEMENTARY?

There are interesting results of the form $PCP[a(n), b(n)] = \texttt{SOMECLASS(n)}$ for multiple classes in the exponential hierarchy: the most famous one is probably $PCP[O(log(n)), O(1)] = NP$. Are ...
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### hardness of approximating clique: how using FGLSS reduction with PCP verifier of hastad

I try to understand the $n^{1-\epsilon}$ hardness of approximating clique for any $\epsilon$ provided in : www.nada.kth.se/~johanh/cliqueinap.ps In fact, I only want to understand the proof of ...
114 views

### Universal constant for bivariate testing

In the seminal paper of Polishchuk and Spielman where they give a construction of nearly linear sized $PCP$ for an $NP$ problem, one of the key ingredients is a low-degree test for bivariate ...
105 views

### Low-degree testing in PCP Theorem using bivariate polynomials

I read about modifications of the low-degree test used in the (first) proof of the PCP theorem. The test used in the proof works over randomly chosen lines while modifications allow choosing random ...
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### Testing - Correcting Pairs in PCPs

The BLR linearity test and the low degree test are two common tools in PCPs. By my understanding these tests ensure bounds such that (self-) correctors can be applied. I have two questions regarding ...
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### A purely graph-theoretic explanation of the reduction from Unique Label Cover to Max-Cut

I am studying the Unique Games Conjecture and the famous reduction to Max-Cut of Khot et al. From their paper and elsewhere on the internet, most authors use (what to me is) an implicit equivalence ...
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### If SAT is in PCP, for some constant q, then P = NP

I have seen this statement before, but I haven't really seen a proof of it: If $SAT\in PCP_{1,2^{−q}}[\log(n),q]$, for some constant $q$, then $P = NP$. Now, if $SAT\in PCP_{1,2^{−q}}[\log(n),q]$, ...
174 views

### Can NP-hard statements be proved by PCPs that only involve reading 2 bits?

For non-negative integers q, let PCP(q) denote the set of promise problems that have polynomial-length probabalistically checkable proofs over the binary alphabet in which the verifier only reads q ...
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### From CHSH inequality to CHSH game

I have been going through Certifiable quantum dice: or, true random number generation secure against quantum adversaries by Umesh Vazirani and Thomas Vidick. They have used entangled particles as ...
158 views

### How are PCPs and ZKPs related?

I only have a (very) introductory knowledge about the Hardness of Approximation and PCP theorem, and I am wondering if it has any specific implications (or can somehow be studied) with Zero Knowledge ...
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### Bivariate low-degree polynomial testing of Polishchuk-Spielman

In the seminal paper of Polishchuk and Spielman where they give a construction of nearly linear sized $PCP$ for an $NP$ problem, one of the key ingredients is a low-degree test for bivariate ...
179 views

### Is a software implementation of a PCP encoder available?

We all know the PCP Theorem. Is there any software package availalbe taking a CNF in e.g. DIMACS format as input, and producing a PCP encoding in the same format as output? It might be interesting to ...
407 views

### Multi prover, verifier games and PCP theorem

This question came up while I was going through Siu On Chan's paper on Approximation Resistance. My question is not really related to the paper though. I also guess that this is more of a reference ...
234 views

### number of PCP queries

we know from the PCP theorem that $PCP[O(log(n)),O(1)]=NP$,what if we choose specific number of queries will the theorem hold ?
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### How to start studying topics Hardness of approximation and PCP's

Recently I have done an introductory course on complexity theory ( which covered 90% of sipser text book). Now I would like to study the topics Hardness of approximation and PCP's. Can you please ...
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### Does Dinur's proof of PCP Theorem imply a procedure for reconstructing a witness?

In Section 3.2 of On Syntactic versus Computational Views on Approximability by Khanna, et al., the authors state that an adaptation of the results from Proof Verification and Hardness of ...
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### What is the query and randomness complexity for very efficient PCPs?

In the 2012 paper On the Concrete-Efficiency Threshold of Probabilistically-Checkable Proofs, the authors state the following (paraphrased from page 11). Theorem 1 (informal). There is a PCP system ...
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### Hardness of approximation without the PCP theorem

An important application of the PCP theorem is that it yields "hardness of approximation" type results. In some relatively simpler cases one can prove such hardness without PCP. Is there, however, any ...
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### $\mathcal{MA}$ in terms of $\mathcal{PCP}$

The probabilistic proof system $\mathcal{PCP}[f(n),g(n)]$ is commonly referred to as a restriction of $\mathcal{MA}$, where Arthur can only use $f(n)$ random bits and can only examine $g(n)$ bits of ...
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### More legent proof of MIP=NEXP using the PCP theorem

Can we prove $\mathsf{MIP}=\mathsf{NEXP}$ using the PCP theorem $\mathsf{NP}=\mathsf{PCP(log(n),O(1))}$ as a shortcut? $\mathsf{MIP}$ is the class of languages with multi-prover interactive proof ...
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### Do good PCPs for NP give us good PCPs for the entire polynomial hierarchy?

The PCP Theorem states that every decision problem in NP has probabilistically checkable proofs (or equivalently, that there exists a complete and quasi-sound proof system for theorems in NP using ...
1k views

### Super-polynomial time approximation algorithms for MAX 3SAT

The PCP theorem states that there is no polynomial time algorithm for MAX 3SAT to find an assignment satisfying $7/8+ \epsilon$ clauses of a satisfiable 3SAT formula unless $P = NP$. There is a ...
407 views

### On the need for a self-correcting function in the PCP theorem

Original proof of the PCP theorem, uses self-correction property of linear functions. Assume we have $f: \{0,1\}^n \rightarrow \{0,1\}$, a function or table of values, that is $(1-\delta)$-close to ...
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### Is there a gap amplification type of result for the Graph Isomorphism Problem?

Suppose $G_1$ and $G_2$ are two undirected graphs on vertex set $\{1, \dotsc, n\}$. The graphs are isomorphic if and only if there is a permutation $\Pi$ such that $G_1 = \Pi(G_2)$, or more formally, ...
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### Effect of serial repetition on soundness of a PCP, and what is special with 1/2?

As far as I know, following operations convert a $PCP_{1,s}[O(\log n),O(1)]$ , to a $PCP_{1,s’}[O(\log n),O(1)]$, with following $s’$ : By constant number of applications of serial repetition: can ...
179 views

### hardness of approximation result for a Min-CSP, by reduction from PCPs

Reduction from PCPs allow us to prove hardness of approximation results for a number of constraint satisfaction problems. I've seen such a reductions only for Max-CSPs. Is this possible only for Max-...
327 views

### Approximating Random MAX-k-SAT

It is known [de la Vega & Karpinski 2002] that random instances of MAX-3-SAT on $n$ variables can be approximated up to fraction at least 8/9 w.h.p. tending to 1 as $n$ tends to infinity. Should ...
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In the degree reduction step of Dinur's proof, the input graph $G$ is transformed into a graph $G'$ by replacing each vertex $v \in V(G)$ by a set of vertices, $cloud(v)$, such that $|cloud(v)| = ... 2answers 790 views ### Is there a simple argument that shows that the unique games conjecture implies the PCP theorem how can one show that what is relation between "Unique games conjecture" and "PCP theorem"? how does one explain "Unique games conjecture" is stronger form of "PCP theorem"? 1answer 361 views ### Alphabet Reduction Step in PCP Proof I understand that the purpose of the alphabet reduction step in Dinur's proof of the PCP theorem is to reduce the alphabet after the graph powering stage. However, I don't see why the alphabet needs ... 1answer 547 views ### PCP Theorem - Alphabet Reduction Step What follows might seem stupid (and that probably reflects my poor understanding - so please bear with me) I had a query on PCP theorem. We know that after the first three steps viz. Degree Reduction,... 2answers 366 views ### PCPs with imperfect completeness The traditional definition of PCPs have perfect completeness -- If$x\in L$, then the prover can give a proof on which the verifier (on reading constantly many bits) always accepts. Suppose we modify ... 2answers 670 views ### Quantum PCP and hardness of simulating of Hamiltonians I have a few questions about Quantum PCP conjecture: What is the statement of the quantum PCP conjecture? What implications would Quantum PCP theorem have for simulating of Hamiltonians? Is it ... 3answers 495 views ### Hard gaps in maximum constraint satisfaction problems? An equivalent formulation of PCP theorem is: For Max 3-SAT it is$NP$-hard to distinguish between satisfiable formulas and formulas where at most$r$-fraction of the clauses are satisfiable (for some$...
It is known that if $P=NP$ then $CoNP= PCP[O(log(n)),O(1)]$. Also, it is known that $NEXP=PCP[poly(n),poly(n)]$. It appears that PCP can't tell us which natural problems are not in $NP$. I wonder if ...