TY - GEN
T1 - Lower bounds on the query complexity of non-uniform and adaptive reductions showing hardness amplification
AU - Artemenko, Sergei
AU - Shaltiel, Ronen
PY - 2011
Y1 - 2011
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=80052359434&partnerID=8YFLogxK
U2 - https://doi.org/10.1007/978-3-642-22935-0_32
DO - https://doi.org/10.1007/978-3-642-22935-0_32
M3 - Conference contribution
SN - 9783642229343
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 377
EP - 388
BT - Approximation, Randomization, and Combinatorial Optimization
T2 - 14th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2011 and the 15th International Workshop on Randomization and Computation, RANDOM 2011
Y2 - 17 August 2011 through 19 August 2011
ER -