Research output per year
Research output per year
Sergei Artemenko, Ronen Shaltiel
Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review
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, R^{D} 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 language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization |
Subtitle of host publication | Algorithms and Techniques - 14th International Workshop, APPROX 2011 and 15th International Workshop, RANDOM 2011, Proceedings |
Pages | 377-388 |
Number of pages | 12 |
DOIs | |
State | Published - 2011 |
Event | 14th 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 2011 → 19 Aug 2011 |
Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 6845 LNCS |
ISSN (Print) | 0302-9743 |
ISSN (Electronic) | 1611-3349 |
Conference | 14th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2011 and the 15th International Workshop on Randomization and Computation, RANDOM 2011 |
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Country/Territory | United States |
City | Princeton, NJ |
Period | 17/08/11 → 19/08/11 |
Research output: Contribution to journal › Article › peer-review