Results 1  10
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14
List Decoding: Algorithms and Applications
 SIGACT News
, 2000
"... Over the years coding theory and complexity theory have benefited from a number of mutually enriching connections. This article focuses on a new connection that has emerged between the two topics in the recent years. This connection is centered around the notion of "listdecoding" for ..."
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Over the years coding theory and complexity theory have benefited from a number of mutually enriching connections. This article focuses on a new connection that has emerged between the two topics in the recent years. This connection is centered around the notion of "listdecoding" for errorcorrecting codes. In this survey we describe the listdecoding problem, the algorithms that have been developed, and a diverse collection of applications within complexity theory. 1
Proofs, Codes, and PolynomialTime Reducibilities
"... We show how to construct proof systems for NP languages where a deterministic polynomialtime verifier can check membership, given any N (2=3)+ffl bits of an N bit witness of membership. ..."
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We show how to construct proof systems for NP languages where a deterministic polynomialtime verifier can check membership, given any N (2=3)+ffl bits of an N bit witness of membership.
The Communication Complexity of Enumeration, Elimination, and Selection
"... Let k, n ∈ N and f: {0, 1} n × {0, 1} n → {0, 1}. Assume Alice has x1,..., xk ∈ {0, 1} n, Bob has y1,..., yk ∈ {0, 1} n, and they want to compute f k (x1x2 · · · xk, y1y2 · · · yk) = (f(x1, y1), · · · , f(xk, yk)) (henceforth f(x1, y1) · · · f(xk, yk)) communicating as few bits as possibl ..."
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Let k, n ∈ N and f: {0, 1} n × {0, 1} n → {0, 1}. Assume Alice has x1,..., xk ∈ {0, 1} n, Bob has y1,..., yk ∈ {0, 1} n, and they want to compute f k (x1x2 · · · xk, y1y2 · · · yk) = (f(x1, y1), · · · , f(xk, yk)) (henceforth f(x1, y1) · · · f(xk, yk)) communicating as few bits as possible. The Direct Sum Conjecture (henceforth DSC) of Karchmer, Raz, and Wigderson, states that the obvious way to compute it (computing f(x1, y1), then f(x2, y2), etc.) is, roughly speaking, the best. This conjecture arose in the study of circuits since a variant of it implies NC 1 � = NC 2. We consider two related problems. Enumeration: Alice and Bob output e ≤ 2 k − 1 elements of {0, 1} k, one of which is f(x1, y1) · · · f(xk, yk). Elimination: Alice and Bob output � b such that � b � = f(x1, y1) · · · f(xk, yk). Selection: (k = 2) Alice and Bob output i ∈ {1, 2} such that if f(x1, y1) = 1 ∨ f(x2, y2) = 1 then f(xi, yi) = 1. a) We devise the Enumeration Conjecture (henceforth ENC) and the Elimination
Algebraic properties for selector functions
 SIAM JOURNAL ON COMPUTING
, 2005
"... The nondeterministic advice complexity of the Pselective sets is known to be exactly linear. Regarding the deterministic advice complexity of the Pselective sets—i.e., the amount of Karp– Lipton advice needed for polynomialtime machines to recognize them in general—the best current upper bound is ..."
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The nondeterministic advice complexity of the Pselective sets is known to be exactly linear. Regarding the deterministic advice complexity of the Pselective sets—i.e., the amount of Karp– Lipton advice needed for polynomialtime machines to recognize them in general—the best current upper bound is quadratic [Ko83] and the best current lower bound is linear [HT96]. We prove that every associatively Pselective set is commutatively, associatively Pselective. Using this, we establish an algebraic sufficient condition for the Pselective sets to have a linear upper bound (which thus would match the existing lower bound) on their deterministic advice complexity: If all Pselective sets are associatively Pselective then the deterministic advice complexity of the Pselective sets is linear. The weakest previously known sufficient condition was P = NP. We also establish related results for algebraic properties of, and advice complexity of, the nondeterministically selective sets.
On the Reducibility of Sets Inside NP to Sets with Low Information Content
, 2002
"... We study whether sets inside NP can be reduced to sets with low information content but possibly still high computational complexity. Examples of sets with low information content are tally sets, sparse sets, Pselective sets and membership comparable sets. For the graph automorphism... ..."
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We study whether sets inside NP can be reduced to sets with low information content but possibly still high computational complexity. Examples of sets with low information content are tally sets, sparse sets, Pselective sets and membership comparable sets. For the graph automorphism...
Sparse Sets, Approximable Sets, and Parallel Queries to NP
 Proc. Sixteenth Symposium on Theoretical Aspects of Computing (STACS '99), LNCS 1563
, 1999
"... We show that if an NPcomplete set or a coNPcomplete set is polynomialtime disjunctive truthtable reducible to a sparse set then FP NP jj = FP NP [log]. With a similar argument we show also that if SAT is O(log n)approximable then FP NP jj = FP NP [log]. Since FP NP jj = FP NP [lo ..."
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We show that if an NPcomplete set or a coNPcomplete set is polynomialtime disjunctive truthtable reducible to a sparse set then FP NP jj = FP NP [log]. With a similar argument we show also that if SAT is O(log n)approximable then FP NP jj = FP NP [log]. Since FP NP jj = FP NP [log] implies that SAT is O(logn)approximable [BFT97], it follows from our result that the two hypotheses are equivalent. We also show that if an NPcomplete set or a coNPcomplete set is disjunctively reducible to a sparse set of polylogarithmic density then P = NP. 1 Introduction The study of the existence of sparse hard sets for complexity classes has occupied complexity theorists for over two decades. The first results in this area were motivated by the BermanHartmanis isomorphism conjecture [BH77] and by the study of connections between uniform and nonuniform complexity classes [KL80]. The focus shifted to proving, for various reducibilities 1 (whose strengths lie between the many...
Membership Comparable and pselective Sets
"... We show that there exists a 2membership comparable set that is not bttreducible to any pselective set. This is a rare example of an unconditional separation in computational complexity. ..."
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We show that there exists a 2membership comparable set that is not bttreducible to any pselective set. This is a rare example of an unconditional separation in computational complexity.
Alternative notions of approximation and spacebounded computations
, 2003
"... We investigate alternative notions of approximation for problems inside P (deterministic polynomial time), and show that even a slightly nontrivial information about a problem may be as hard to obtain as the solution itself. For example, we prove that if one could eliminate even a single possibilit ..."
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We investigate alternative notions of approximation for problems inside P (deterministic polynomial time), and show that even a slightly nontrivial information about a problem may be as hard to obtain as the solution itself. For example, we prove that if one could eliminate even a single possibility for the value of an arithmetic circuit on a given input, then this would imply that the class P has fast (polygarithmic time) parallel solutions. In other words, this would constitute a proof that there are no inherently sequential problems in P, which is quite unlikely. The result is robust with respect to eliminating procedures that are allowed to err (by excluding the correct value) with small probability. We also show that several fundamental linear algebra problems are hard in this sense. It turns out that it is as hard to substantially reduce the number of possible values for the determinant and rank as to compute them exactly. Finally, we show that (in some precise sense) randomness can be nontrivially substituted for nondeterminism in space. Although it is believed that randomness does not give more than a constant factor advantage in space over determinism, it is not even known whether it is no more powerful than nondeterminism. We will show that the latter is true for a restricted version of probabilistic logspace, where the error is potentially larger than what can be achieved by amplification.
SIGACT News Complexity Theory Column 25
"... The 2000s are upon us, and your Complexity Theory Column celebrates their start with Madhu Sudan writing on list decoding. Up next issue is Ulrich Hertrampf writing on algebraic acceptance ..."
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The 2000s are upon us, and your Complexity Theory Column celebrates their start with Madhu Sudan writing on list decoding. Up next issue is Ulrich Hertrampf writing on algebraic acceptance