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15
A new multilayered PCP and the hardness of hypergraph vertex cover
 In Proceedings of the 35th Annual ACM Symposium on Theory of Computing
, 2003
"... Abstract Given a kuniform hypergraph, the EkVertexCover problem is to find the smallest subsetof vertices that intersects every hyperedge. We present a new multilayered PCP construction that extends the Raz verifier. This enables us to prove that EkVertexCover is NPhard toapproximate within a ..."
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Cited by 53 (10 self)
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Abstract Given a kuniform hypergraph, the EkVertexCover problem is to find the smallest subsetof vertices that intersects every hyperedge. We present a new multilayered PCP construction that extends the Raz verifier. This enables us to prove that EkVertexCover is NPhard toapproximate within a factor of ( k 1 ") for arbitrary constants "> 0 and k> = 3. The resultis nearly tight as this problem can be easily approximated within factor k. Our constructionmakes use of the biased LongCode and is analyzed using combinatorial properties of swise tintersecting families of subsets.We also give a different proof that shows an inapproximability factor of b k 2 c ". In additionto being simpler, this proof also works for superconstant values of k up to (log N)1/c where
Inapproximability of combinatorial optimization problems
 Electronic Colloquium on Computational Complexity
, 2004
"... ..."
Query efficient PCPs with perfect completeness
 In 42nd Annual Symposium on Foundations of Computer Science
, 2001
"... For every integer k > 0, we present a PCP characterization of NP where the verifier uses logarithmic randomness, queries 4k + k 2 bits in the proof, accepts a correct proof with probability 1 (i.e. it is has perfect completeness) and accepts any supposed proof of a false statement with probability ..."
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Cited by 15 (3 self)
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For every integer k > 0, we present a PCP characterization of NP where the verifier uses logarithmic randomness, queries 4k + k 2 bits in the proof, accepts a correct proof with probability 1 (i.e. it is has perfect completeness) and accepts any supposed proof of a false statement with probability at most 2 k 2 +1 . In particular, the verifier achieves optimal amortized query complexity of 1 + for arbitrarily small constant > 0. Such a characterization was already proved by Samorodnitsky and Trevisan [15], but their verifier loses perfect completeness and their proof makes an essential use of this feature. By using an adaptive verifier we can decrease the number of query bits to 2k + k 2 , the same number obtained in [15]. Finally we extend some of the results to larger domains. Royal Institute of Technology, Stockholm, work done while visiting Institute for Advanced Study, supported by NSF grant CCR9987077.
The Complexity of Properly Learning Simple Concept Classes
, 2007
"... We consider the complexity of properly learning concept classes, i.e. when the learner must output a hypothesis of the same form as the unknown concept. We present the following new upper and lower bounds on wellknown concept classes: • We show that unless NP = RP, there is no polynomialtime PAC l ..."
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Cited by 9 (1 self)
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We consider the complexity of properly learning concept classes, i.e. when the learner must output a hypothesis of the same form as the unknown concept. We present the following new upper and lower bounds on wellknown concept classes: • We show that unless NP = RP, there is no polynomialtime PAC learning algorithm for DNF formulas where the hypothesis is an ORofthresholds. Note that as special cases, we show that neither DNF nor ORofthresholds are properly learnable unless NP = RP. Previous hardness results have required strong restrictions on the size of the output DNF formula. We also prove that it is NPhard to learn the intersection of ℓ ≥ 2 halfspaces by the intersection of k halfspaces for any constant k ≥ 0. Previous work held for the case when k = ℓ. • Assuming that NP � ⊆ DTIME(2nɛ) for a certain constant ɛ < 1 we show that it is not possible to learn size s decision trees by size sk decision trees for any k ≥ 0. Previous hardness results for learning decision trees held for k ≤ 2. • We present the first nontrivial upper bounds on properly learning DNF formulas. More specifically, we show how to learn size s DNF by DNF in time 2 Õ( √ n log s). The hardness results for DNF formulas and intersections of halfspaces are obtained via specialized
Is Constraint Satisfaction Over Two Variables Always Easy?
 Algorithms
, 2002
"... By the breakthrough work of Hstad, several constraint satisfaction problems are now known to have the following approximation resistance property: satisfying more clauses than what picking a random assignment would achieve is NPhard. This is the case for example for Max E3Sat, Max E3Lin and Max E ..."
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Cited by 7 (2 self)
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By the breakthrough work of Hstad, several constraint satisfaction problems are now known to have the following approximation resistance property: satisfying more clauses than what picking a random assignment would achieve is NPhard. This is the case for example for Max E3Sat, Max E3Lin and Max E4Set Splitting. A notable exception to this extreme hardness is constraint satisfaction over two variables (2CSP); as a corollary of the celebrated GoemansWilliamson algorithm, we know that every Boolean 2CSP has a nontrivial approximation algorithm whose performance ratio is better than that obtained by picking a random assignment to the variables. An intriguing question then is whether this is also the case for 2CSPs over larger, nonBoolean domains. This question is still open, and is equivalent to whether the generalization of Max 2SAT to domains of size d, can be approximated to a factor better than (1 ).
Hardness of approximating the closest vector problem with preprocessing
 In FOCS
, 2005
"... Abstract We show that, unless NP`DTIME(2poly log(n)), the closest vector problem with preprocessing, for `p norm forany p> = 1, is hard to approximate within a factor of(log n)1/pffl for any ffl> 0. This improves the previous bestfactor of 3 1/p ffl due to Regev [19]. Our results also imply that ..."
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Cited by 6 (0 self)
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Abstract We show that, unless NP`DTIME(2poly log(n)), the closest vector problem with preprocessing, for `p norm forany p> = 1, is hard to approximate within a factor of(log n)1/pffl for any ffl> 0. This improves the previous bestfactor of 3 1/p ffl due to Regev [19]. Our results also imply that under the same complexity assumption, the nearestcodeword problem with preprocessing is hard to approximate within a factor of (log n)1ffl for any ffl> 0.
Hardness of max 3SAT with no mixed clauses
 In IEEE Conference on Computational Complexity
, 2005
"... We study the complexity of approximating Max NME3SAT, a variant of Max 3SAT when the instances are guaranteed to not have any mixed clauses, i.e., every clause has either all its literals unnegated or all of them negated. This is a natural special case of Max 3SAT introduced in [7], where the quest ..."
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Cited by 4 (0 self)
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We study the complexity of approximating Max NME3SAT, a variant of Max 3SAT when the instances are guaranteed to not have any mixed clauses, i.e., every clause has either all its literals unnegated or all of them negated. This is a natural special case of Max 3SAT introduced in [7], where the question of whether this variant can be approximated within a factor better than 7/8 was also posed. We prove that it is NPhard to approximate Max NME3SAT within a factor of 7/8 + ε for arbitrary ε> 0, and thus this variant is no easier to approximate than general Max 3SAT. The proof uses the technique of multilayered PCPs, introduced in [3], to avoid the technical requirement of folding of the proof tables. Circumventing this requirement means that the PCP verifier can use the bits it accesses without additional negations, and this leads to a hardness for Max 3SAT without any mixed clauses. 1
Bypassing UGC from some Optimal Geometric Inapproximability Results
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 177
, 2010
"... The Unique Games conjecture (UGC) has emerged in recent years as the starting point for several optimal inapproximability results. While for none of these results a reverse reduction to Unique Games is known, the assumption of bijective projections in the Label Cover instance seems critical in these ..."
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The Unique Games conjecture (UGC) has emerged in recent years as the starting point for several optimal inapproximability results. While for none of these results a reverse reduction to Unique Games is known, the assumption of bijective projections in the Label Cover instance seems critical in these proofs. In this work we bypass the UGC assumption in inapproximability results for two geometric problems, obtaining a tight NPhardness result in each case. The first problem known as the Lp Subspace Approximation is a generalization of the classic least squares regression problem. Here, the input consists of a set of points S = {a1,..., am} ⊆ R n and a parameter k (possibly depending on n). The goal is to find a subspace H of R n of dimension k that minimizes the sum of the p th powers of the distances to the points. For p = 2, k = n − 1, this reduces to the least squares regression problem, while for p = ∞, k = 0 it reduces to the problem of finding a ball of minimum radius enclosing all the points. We show that for any fixed p (2 < p < ∞) it is NPhard to approximate this problem to within a factor of γp − ɛ for constant ɛ> 0, where γp is the pth moment of a standard Gaussian variable. This matches the factor γp approximation algorithm obtained by Deshpande, Tulsiani and Vishnoi
Circumventing dto1 for Approximation Resistance of Satisfiable Predicates Strictly Containing Parity of Width at Least Four
, 2013
"... Håstad established that any predicate P ⊆ {0,1} m containing Parity of width at least three is approximation resistant for almostsatisfiable instances. In comparison to for example the approximation hardness of 3SAT, this general result however left open the hardness of perfectlysatisfiable insta ..."
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Håstad established that any predicate P ⊆ {0,1} m containing Parity of width at least three is approximation resistant for almostsatisfiable instances. In comparison to for example the approximation hardness of 3SAT, this general result however left open the hardness of perfectlysatisfiable instances. This limitation was addressed by O’Donnell and Wu, and subsequently generalized by Huang, to show the threshold result that predicates strictly containing Parity of width at least three are approximation resistant also for perfectlysatisfiable instances, assuming the dto1 Conjecture. We extend modern hardnessofapproximation techniques by Mossel et al., eliminating the dependency on projection degrees for a special case of decoupling/invariance and—when
On the NPhardness of MaxNot2
, 2012
"... We prove that, for any ɛ> 0, it is NPhard to, given a satisfiable instance of MaxNTW (Not2), find an assignment that satisfies a fraction 5 + ɛ of the constraints. This, up to the existence of ɛ, matches the 8 approximation ratio obtained by the trivial algorithm that just picks an assignment at ..."
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We prove that, for any ɛ> 0, it is NPhard to, given a satisfiable instance of MaxNTW (Not2), find an assignment that satisfies a fraction 5 + ɛ of the constraints. This, up to the existence of ɛ, matches the 8 approximation ratio obtained by the trivial algorithm that just picks an assignment at random and thus the result is tight. Said equivalently the result proves that MaxNTW is approximation resistant on satisfiable instances and this makes our understanding of arity three MaxCSPs with regards to approximation resistance complete.