arXiv 2011.023202020-11cs.CCcs.CL

PCP定理、SETH及更多：证明次线性时间不可近似性

PCP Theorems, SETH and More: Towards Proving Sub-linear Time Inapproximability

Hengzhao Ma, Jianzhong Li

备注：: This paper is an old version of another paper submitted to arxiv, with id 2107.01520. Moreover, this paper contains mistakes. Theorem 5.1 in this paper is wrong. The reason is that Merlin can make Alice believe q\in S by sending another q'\ne q but q'\in S. Another player Bob should be included in the communication protocol to avoid this situation. The paper 2107.01520 fixed this problem

摘要：

在本文中，我们提出了次线性时间不可逼近的PCP类定理。Abboud等人设计了次二次时间不可逼近的分布式PCP框架。我们证明了分布式PCP定理可以推广到证明任意多项式时间的不可逼近性，但在线性情况下失败。我们证明了次线性PCP定理适应从MA协议的集合包含问题，并展示了如何使用该定理证明现有的和新的不可逼近的结果，展示了次线性PCP定理的力量。考虑到近年来关于次线性时间算法的研究工作，次线性PCP定理对于指导次线性时间近似算法的研究具有重要意义。

英文摘要：

In this paper we propose the PCP-like theorem for sub-linear time inapproximability. Abboud et al. have devised the distributed PCP framework for sub-quadratic time inapproximability. We show that the distributed PCP theorem can be generalized for proving arbitrary polynomial time inapproximability, but fails in the linear case. We prove the sub-linear PCP theorem by adapting from an MA-protocol for the Set Containment problem, and show how to use the theorem to prove both existing and new inapproximability results, exhibiting the power of the sub-linear PCP theorem. Considering the emerging research works on sub-linear time algorithms, the sub-linear PCP theorem is important in guiding the research in sub-linear time approximation algorithms.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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