ECCC-Report TR19-023https://eccc.weizmann.ac.il/report/2019/023Comments and Revisions published for TR19-023en-usFri, 15 Nov 2019 23:44:02 +0200
Revision 3
| Smooth and Strong PCPs |
Orr Paradise
https://eccc.weizmann.ac.il/report/2019/023#revision3Probabilistically checkable proofs (PCPs) can be verified based only on a constant amount of random queries, such that any correct claim has a proof that is always accepted, and incorrect claims are rejected with high probability (regardless of the given alleged proof). We consider two possible features of PCPs:
- A PCP is *strong* if it rejects an alleged proof of a correct claim with probability proportional to its distance from some correct proof of that claim.
- A PCP is *smooth* if each location in a proof is queried with equal probability.
We prove that all sets in $\mathcal{NP}$ have a smooth and strong PCP of polynomial length that can be verified based on a constant number of queries. We do so by following the proof of the PCP theorem of Arora, Lund, Motwani, Sudan and Szegedy (JACM, 1998), providing a stronger analysis of the Hadamard and Reed--Muller based PCPs and a refined PCP composition theorem. In fact, we show that any set in $\mathcal{NP}$ has a smooth strong *canonical* PCP of Proximity (PCPP), meaning that there is an efficiently computable bijection of $\mathcal{NP}$ witnesses to correct proofs.
This improves on the recent result of Dinur, Gur and Goldreich (ITCS, 2019) that constructs strong canonical PCPPs that are inherently non-smooth. Our result implies the hardness of approximating the satisfiability of "stable" 3CNF formulae with bounded variable occurrence, proving a hypothesis used in the work of Friggstad, Khodamoradi and Salavatipour (SODA, 2019). Here *stability* means that the number of clauses violated by an assignment is proportional to its distance from a satisfying assignment (in the relative Hamming metric).Fri, 15 Nov 2019 23:44:02 +0200https://eccc.weizmann.ac.il/report/2019/023#revision3
Revision 2
| Smooth and Strong PCPs |
Orr Paradise
https://eccc.weizmann.ac.il/report/2019/023#revision2Probabilistically checkable proofs (PCPs) can be verified based only on a constant amount of random queries, such that any correct claim has a proof that is always accepted, and incorrect claims are rejected with high probability (regardless of the given alleged proof). We consider two possible features of PCPs:
- A PCP is *strong* if it rejects an alleged proof of a correct claim with probability proportional to its distance from some correct proof of that claim.
- A PCP is *smooth* if each location in a proof is queried with equal probability.
We prove that all sets in NP have PCPs that are both smooth and strong, are of polynomial length, and can be verified based on a constant number of queries. This is achieved by following the proof of the PCP theorem of Arora, Lund, Motwani, Sudan and Szegedy (JACM, 1998), providing a stronger analysis of the Hadamard and Reed--Muller based PCPs and a refined PCP composition theorem. In fact, we show that any set in NP has a smooth strong *canonical* PCP of Proximity (PCPP), meaning that there is an efficiently computable bijection of NP witnesses to correct proofs. This improves on the recent construction of Dinur, Gur and Goldreich (ITCS, 2019) of PCPPs that are strong canonical but inherently non-smooth.
Our result implies the hardness of approximating the satisfiability of "stable" 3CNF formulae with bounded variable occurrence, where stable means that the number of clauses violated by an assignment is proportional to its distance from a satisfying assignment (in the relative Hamming metric). This proves a hypothesis used in the work of Friggstad, Khodamoradi and Salavatipour (SODA, 2019), suggesting a connection between the hardness of these instances and other stable optimization problems.Fri, 15 Nov 2019 23:39:53 +0200https://eccc.weizmann.ac.il/report/2019/023#revision2
Revision 1
| Smooth and Strong PCPs |
Orr Paradise
https://eccc.weizmann.ac.il/report/2019/023#revision1Probabilistically checkable proofs (PCPs) can be verified based only on a constant amount of random queries, such that any correct claim has a proof that is always accepted, and incorrect claims are rejected with high probability (regardless of the given alleged proof). We consider two possible features of PCPs:
* A PCP is *strong* if it rejects an alleged proof of a correct claim with probability proportional to its distance from some correct proof of that claim.
* A PCP is *smooth* if each location in a proof is queried with equal probability.
We prove that all sets in NP have PCPs that are both smooth and strong, are of polynomial length, and can be verified based on a constant number of queries. This is achieved by following the proof of the PCP theorem of Arora, Lund, Motwani, Sudan and Szegedy (JACM, 1998), providing a stronger analysis of the Hadamard and Reed--Muller based PCPs and a refined PCP composition theorem. In fact, we show that any set in NP has a smooth strong *canonical* PCP of Proximity (PCPP), meaning that there is an efficiently computable bijection of NP witnesses to correct proofs. This improves on the recent result of Dinur, Gur and Goldreich (ITCS, 2019) that constructs strong canonical PCPPs that are inherently non-smooth.
Our result implies the hardness of approximating the satisfiability of “stable” 3CNF formulae with bounded variable occurrence, where stable means that the number of clauses violated by an assignment is proportional to its distance from a satisfying assignment (in the relative Hamming metric). This proves a hypothesis used in the work of Friggstad, Khodamoradi and Salavatipour (SODA, 2019), suggesting a connection between the hardness of these instances and other stable optimization problems.Sun, 01 Sep 2019 02:21:43 +0300https://eccc.weizmann.ac.il/report/2019/023#revision1
Paper TR19-023
| Smooth and Strong PCPs |
Orr Paradise
https://eccc.weizmann.ac.il/report/2019/023Probabilistically checkable proofs (PCPs) can be verified based only on a constant amount of random queries, such that any correct claim has a proof that is always accepted, and incorrect claims are rejected with high probability (regardless of the given alleged proof). We consider two possible features of PCPs:
- A PCP is *strong* if it rejects an alleged proof of a correct claim with probability proportional to its distance from some correct proof of that claim.
- A PCP is *smooth* if each location in a proof is queried with equal probability.
We prove that all sets in $\mathcal{NP}$ have a smooth and strong PCP of polynomial length that can be verified based on a constant number of queries. We do so by following the proof of the PCP theorem of Arora, Lund, Motwani, Sudan and Szegedy (JACM, 1998), providing a stronger analysis of the Hadamard and Reed--Muller based PCPs and a refined PCP composition theorem. In fact, we show that any set in $\mathcal{NP}$ has a smooth strong *canonical* PCP of Proximity (PCPP), meaning that there is an efficiently computable bijection of $\mathcal{NP}$ witnesses to correct proofs.
This improves on the recent result of Dinur, Gur and Goldreich (ITCS, 2019) that constructs strong canonical PCPPs that are inherently non-smooth. Our result implies the hardness of approximating the satisfiability of "stable" 3CNF formulae with bounded variable occurrence, proving a hypothesis used in the work of Friggstad, Khodamoradi and Salavatipour (SODA, 2019). Here *stability* means that the number of clauses violated by an assignment is proportional to its distance from a satisfying assignment (in the relative Hamming metric).Tue, 26 Feb 2019 00:38:50 +0200https://eccc.weizmann.ac.il/report/2019/023