Lower bounds on the query complexity of non-uniform and adaptive reductions showing hardness amplification

Sergei Artemenko, Ronen Shaltiel

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Hardness amplification results show that for every function f there exists a function Amp(f) such that the following holds: if every circuit of size s computes f correctly on at most a 1 - δ fraction of inputs, then every circuit of size s′ computes Amp(f) correctly on at most a 1/2 + ε fraction of inputs. All hardness amplification results in the literature suffer from "size loss" meaning that s′ ≤ ε·s. In this paper we show that proofs using "non-uniform reductions" must suffer from size loss. To the best of our knowledge, all proofs in the literature are by non-uniform reductions. Our result is the first lower bound that applies to non-uniform reductions that are adaptive. A reduction is an oracle circuit R(•) such that when given oracle access to any function D that computes Amp(f) correctly on a 1/2 + ε fraction of inputs, RD computes f correctly on a 1 - δ fraction of inputs. A non-uniform reduction is allowed to also receive a short advice string α that may depend on both f and D in an arbitrary way. The well known connection between hardness amplification and list-decodable error-correcting codes implies that reductions showing hardness amplification cannot be uniform for ε < 1/4. A reduction is non-adaptive if it makes non-adaptive queries to its oracle. Shaltiel and Viola (STOC 2008) showed lower bounds on the number of queries made by non-uniform reductions that are non-adaptive. We show that every non-uniform reduction must make at least Ω(1/ε) queries to its oracle (even if the reduction is adaptive). This implies that proofs by non-uniform reductions must suffer from size loss. We also prove the same lower bounds on the number of queries of non-uniform and adaptive reductions that are allowed to rely on arbitrary specific properties of the function f. Previous limitations on reductions were proven for "function-generic" hardness amplification, in which the non-uniform reduction needs to work for every function f and therefore cannot rely on specific properties of the function.

Original languageAmerican English
Title of host publicationApproximation, Randomization, and Combinatorial Optimization
Subtitle of host publicationAlgorithms and Techniques - 14th International Workshop, APPROX 2011 and 15th International Workshop, RANDOM 2011, Proceedings
Pages377-388
Number of pages12
DOIs
StatePublished - 2011
Event14th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2011 and the 15th International Workshop on Randomization and Computation, RANDOM 2011 - Princeton, NJ, United States
Duration: 17 Aug 201119 Aug 2011

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6845 LNCS

Conference

Conference14th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2011 and the 15th International Workshop on Randomization and Computation, RANDOM 2011
Country/TerritoryUnited States
CityPrinceton, NJ
Period17/08/1119/08/11

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • General Computer Science

Fingerprint

Dive into the research topics of 'Lower bounds on the query complexity of non-uniform and adaptive reductions showing hardness amplification'. Together they form a unique fingerprint.

Cite this