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27
Algorithmic construction of sets for krestrictions
 ACM TRANSACTIONS ON ALGORITHMS
, 2006
"... This work addresses krestriction problems, which unify combinatorial problems of the following type: The goal is to construct a short list of strings in Σ m that satisfies a given set of kwise demands. For every k positions and every demand, there must be at least one string in the list that satis ..."
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Cited by 73 (2 self)
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This work addresses krestriction problems, which unify combinatorial problems of the following type: The goal is to construct a short list of strings in Σ m that satisfies a given set of kwise demands. For every k positions and every demand, there must be at least one string in the list that satisfies the demand at these positions. Problems of this form frequently arise in different fields in Computer Science. The standard approach for deterministically solving such problems is via almost kwise independence or kwise approximations for other distributions. We offer a generic algorithmic method that yields considerably smaller constructions. To this end, we generalize a previous work of Naor, Schulman and Srinivasan [18]. Among other results, we greatly enhance the combinatorial objects in the heart of their method, called splitters, and construct multiway splitters, using a new discrete version of the topological Necklace Splitting Theorem [1]. We utilize our methods to show improved constructions for group testing [19] and generalized hashing [3], and an improved inapproximability result for SetCover under the assumption P != NP.
Two Query PCP with SubConstant Error
, 2008
"... We show that the N PComplete language 3SAT has a PCP verifier that makes two queries to a proof of almostlinear size and achieves subconstant probability of error o(1). The verifier performs only projection tests, meaning that the answer to the first query determines at most one accepting answer ..."
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Cited by 51 (5 self)
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We show that the N PComplete language 3SAT has a PCP verifier that makes two queries to a proof of almostlinear size and achieves subconstant probability of error o(1). The verifier performs only projection tests, meaning that the answer to the first query determines at most one accepting answer to the second query. Previously, by the parallel repetition theorem, there were PCP Theorems with twoquery projection tests, but only (arbitrarily small) constant error and polynomial size [29]. There were also PCP Theorems with subconstant error and almostlinear size, but a constant number of queries that is larger than 2 [26]. As a corollary, we obtain a host of new results. In particular, our theorem improves many of the hardness of approximation results that are proved using the parallel repetition theorem. A partial list includes the following: 1. 3SAT cannot be efficiently approximated to within a factor of 7 8 + o(1), unless P = N P. This holds even under almostlinear reductions. Previously, the best known N Phardness
Simple PCPs with Polylog Rate and Query Complexity
, 2005
"... We give constructions of probabilistically checkable proofs (PCPs) of length n·poly(log n) (to prove satisfiability of circuits of size n) that can verified by querying poly(log n) bits of the proof. We also give constructions of locally testable codes (LTCs) with similar parameters. Previous constr ..."
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Cited by 43 (10 self)
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We give constructions of probabilistically checkable proofs (PCPs) of length n·poly(log n) (to prove satisfiability of circuits of size n) that can verified by querying poly(log n) bits of the proof. We also give constructions of locally testable codes (LTCs) with similar parameters. Previous constructions of short PCPs (from [5] to [9]) relied extensively on properties of low degree multivariate polynomials. In contrast, our constructions rely on new problems and techniques revolving around the properties of codes based on high degree polynomials in one variable (also known as ReedSolomon codes). We show how to convert the problem of verifying the satisfaction of a circuit by a given assignment to the task of verifying that a given function is close to being a ReedSolomon codeword, i.e., a univariate polynomial of specified degree. This reduction is simpler than the corresponding steps in previous reductions, and gives a new alternative to using the popular “sumcheck protocol”. We then give a new PCP for the special task of proving that a function is close to being a ReedSolomon codeword. This step of the construction is by a selfcontained recursion, and the only ingredient needed in the analysis is the bivariate lowdegree test of Polischuk and Spielman [27]. Note that our constructions yield LTCs first, which are then converted to PCPs. In contrast, most recent constructions go in the opposite (and less natural) direction of getting LTCs from PCPs.
Short PCPs with Polylog Query Complexity
 SIAM J. COMPUT. VOL. 38, NO. 2, PP. 551–607
, 2008
"... We give constructions of probabilistically checkable proofs (PCPs) of length n · polylog n proving satisfiability of circuits of size n that can be verified by querying polylog n bits of the proof. We also give analogous constructions of locally testable codes (LTCs) mapping n information bits to n ..."
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Cited by 32 (7 self)
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We give constructions of probabilistically checkable proofs (PCPs) of length n · polylog n proving satisfiability of circuits of size n that can be verified by querying polylog n bits of the proof. We also give analogous constructions of locally testable codes (LTCs) mapping n information bits to n · polylog n bit long codewords that are testable with polylog n queries. Our constructions rely on new techniques revolving around properties of codes based on relatively highdegree polynomials in one variable, i.e., Reed–Solomon codes. In contrast, previous constructions of
Polynomial Flowcut Gaps and Hardness of Directed Cut Problems
 In Proc. of STOC, 2007
"... We study the multicut and the sparsest cut problems in directed graphs. In the multicut problem, we are a given an nvertex graph G along with k sourcesink pairs, and the goal is to find the minimum cardinality subset of edges whose removal separates all sourcesink pairs. The sparsest cut problem ..."
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Cited by 25 (0 self)
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We study the multicut and the sparsest cut problems in directed graphs. In the multicut problem, we are a given an nvertex graph G along with k sourcesink pairs, and the goal is to find the minimum cardinality subset of edges whose removal separates all sourcesink pairs. The sparsest cut problem has the same input, but the goal is to find a subset of edges to delete so as to minimize the ratio of the number of deleted edges to the number of sourcesink pairs that are separated by this deletion. The natural linear programming relaxation for multicut corresponds, by LPduality, to the wellstudied maximum (fractional) multicommodity flow problem, while the standard LPrelaxation for sparsest cut corresponds to maximum concurrent flow. Therefore, the integrality gap of the linear programming relaxation for multicut/sparsest cut is also the flowcut gap: the largest gap, achievable for any graph, between the maximum flow value and the minimum cost solution for the corresponding cut problem. Our first result is that the flowcut gap between maximum multicommodity flow and minimum multicut is ˜ Ω(n 1/7) in directed graphs. We show a similar result for the gap between maximum concurrent flow and sparsest cut in directed graphs. These results improve upon a
SubConstant Error Low Degree Test of Almost Linear Size
 In STOC
, 2006
"... Given a function f: � m → � over a finite field �, a low degree tester tests its agreement with an mvariate polynomial of total degree at most d over �. The tester is usually given access to an oracle A providing the supposed restrictions of f to affine subspaces of constant dimension (e.g., line ..."
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Given a function f: � m → � over a finite field �, a low degree tester tests its agreement with an mvariate polynomial of total degree at most d over �. The tester is usually given access to an oracle A providing the supposed restrictions of f to affine subspaces of constant dimension (e.g., lines, planes, etc.). The tester makes very few (probabilistic) queries to f and to A (say, one query to f and one query to A), and decides whether to accept or reject based on the replies. We wish to minimize two parameters of a tester: its error and its size. The error bounds the probability that the tester accepts although the function is far from a low degree polynomial. The size is the number of bits required to write the oracle replies on all possible tester’s queries. Low degree testing is a central ingredient in most constructions of probabilistically checkable proofs (P CP s) and locally testable codes (LT Cs). The error of the low degree tester is related to the soundness of the P CP and its size is related to the size of the P CP (or the length of the LT C). We design and analyze new low degree testers that have both subconstant error o(1) and almostlinear size n 1+o(1) (where n = �  m). Previous constructions of subconstant error testers had polynomial size [3, 16]. These testers enabled the construction of P CP s with subconstant soundness, but polynomial size [3, 16, 9]. Previous constructions of almostlinear size testers obtained only constant error [13, 7]. These testers were used to construct almostlinear size LT Cs and almostlinear size P CP s with constant soundness
Quantum information and the PCP theorem
 In FOCS
, 2005
"... We show how to encode 2n (classical) bits a1,...,a2 n by a single quantum state Ψ 〉 of size O(n) qubits, such that: for any constant k and any i1,...,ik ∈ {1,...,2 n}, the values of the bits ai1,...,aik can be retrieved from Ψ 〉 by a oneround ArthurMerlin interactive protocol of size polynomial ..."
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Cited by 15 (1 self)
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We show how to encode 2n (classical) bits a1,...,a2 n by a single quantum state Ψ 〉 of size O(n) qubits, such that: for any constant k and any i1,...,ik ∈ {1,...,2 n}, the values of the bits ai1,...,aik can be retrieved from Ψ 〉 by a oneround ArthurMerlin interactive protocol of size polynomial in n. This shows how to go around HolevoNayak’s Theorem, using ArthurMerlin proofs. We use the new representation to prove the following results: 1. Interactive proofs with quantum advice: We show that the class QIP/qpoly contains all languages. That is, for any language L (even nonrecursive), the membership x ∈ L (for x of length n) can be proved by a polynomialsize quantum interactive proof, where the verifier is a polynomialsize quantum circuit with working space initiated with some quantum state ΨL,n〉 (depending only on L and n). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. 2. PCP with only one query: We show that the membership x ∈ SAT (for x of length n) can be proved by a logarithmicsize quantum state Ψ〉, together with a polynomialsize classical proof consisting of blocks of length polylog(n) bits each, such that after measuring the state Ψ 〉 the verifier only needs to read one block of the classical proof. While the first result is a straight forward consequence of the new representation, the second requires an additional machinery of quantum lowdegreetest that may be interesting in its own right.
ApproximatingCVP to Within AlmostPolynomial Factors is NPhard
 In Proc. 39th IEEE Symp. on Foundations of Computer Science
, 1998
"... This paper shows the closest vector in a lattice to be NPhard to approximate to within any factor up to 2 (log n) 1\Gammaffl where ffl = (log log n) \Gammaff for any constant ff ! 1 2 . Introduction Background A lattice L = L(v 1 ; ::; vn ), for vectors v 1 ; ::; vn 2 R n is the set of ..."
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This paper shows the closest vector in a lattice to be NPhard to approximate to within any factor up to 2 (log n) 1\Gammaffl where ffl = (log log n) \Gammaff for any constant ff ! 1 2 . Introduction Background A lattice L = L(v 1 ; ::; vn ), for vectors v 1 ; ::; vn 2 R n is the set of all integer linear combinations of v 1 ; ::; vn , that is, L = f P a i v i j a i 2 Zg. Given a lattice L and an arbitrary vector y, the Closest Vector Problem (CVP) is to find a vector in L closest to y. The Shortest Vector Problem (SVP) is the homogeneous analog of CVP, i.e. finding the shortest nonzero vector in L. These lattice problems have been introduced in the previous century, and have been studied since. Minkowsky and Dirichlet tried, with little success, to come up with lattice approximation algorithms. It was much later that the lattice reduction algorithm was presented by Lenstra, Lenstra and Lov'asz [L L L82] , achieving a polynomialtime algorithm approximating the Shortest...