2
            <sup>
              log1-ε
              <i>n</i>
            </sup>
            hardness for the closest vector problem with preprocessing

Khot, Subhash; Popat, Preyas; Vishnoi, Nisheeth K.

doi:10.1145/2213977.2214004

Cited by 6 publications

(4 citation statements)

References 24 publications

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“…Nearest lattice vector (CVP) when the lattice can be preprocessed was shown to be NP-hard and APX-hard by Feige and Micciancio [10]. The tightest hardness results for lattice problems with preprocessing currently known are by Khot et al [16]. An earlier work by Alekhnovich et al [2] has some partial overlap with our current work, because it uses PCP theory and in the process gives hardness of approximation results with preprocessing for additional problems.…”

Section: Related Workmentioning

confidence: 73%

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Universal Factor Graphs

Feige

Jozeph

2012

Automata, Languages, and Programming

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The factor graph of an instance of a symmetric constraint satisfaction problem on n Boolean variables and m constraints (CSPs such as k-SAT, k-AND, k-LIN) is a bipartite graph describing which variables appear in which constraints. The factor graph describes the instance up to the polarity of the variables, and hence there are up to 2 km instances of the CSP that share the same factor graph. It is well known that factor graphs with certain structural properties make the underlying CSP easier to either solve exactly (e.g., for tree structures) or approximately (e.g., for planar structures). We are interested in the following question: is there a factor graph for which if one can solve every instance of the CSP with this particular factor graph, then one can solve every instance of the CSP regardless of the factor graph (and similarly, for approximation)? We call such a factor graph universal. As one needs different factor graphs for different values of n and m, this gives rise to the notion of a family of universal factor graphs. We initiate a systematic study of universal factor graphs, and present some results for max-kSAT. Our work has connections with the notion of preprocessing as previously studied for closest codeword and closest lattice-vector problems, with proofs for the PCP theorem, and with tests for the long code. Many questions remain open.

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Section: Related Workmentioning

confidence: 73%

“…Quoting from[2]: "The proof of this theorem, which is a laborious and an almost exact mimic of the proof of the PCP Theorem, is beyond the scope of this version of the paper." A subsequent paper[16] that extends[2] no longer uses this theorem, and hence does not contain the proof either.…”

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confidence: 99%

Universal Factor Graphs

Feige

Jozeph

2012

Automata, Languages, and Programming

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“…The CVPP has been investigated in several papers [42,24,55,18,7,38], mostly with the goal of showing that CVP is NP-hard even for fixed families of lattices, or of devising polynomial time approximation algorithms (with superpolynomial time preprocessing). In summary, CVPP is NP-hard to approximate for any constant factors [7] (and quasi NP-hard for certain subpolynomial factors [38]), and it can be approximated in polynomial time within a factor O( n/ log n) [2], at least in its distance estimation variant. In this paper we use CVPP mostly as a building block to give a modular description of our CVP algorithm.…”

Section: Related Workmentioning

confidence: 99%

A Deterministic Single Exponential Time Algorithm for Most Lattice Problems Based on Voronoi Cell Computations

Micciancio¹,

Voulgaris²

2013

SIAM J. Comput.

161

188

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We give deterministicÕ(2 2n )-timeÕ(2 n )-space algorithms to solve all the most important computational problems on point lattices in NP, including the shortest vector problem (SVP), closest vector problem (CVP), and shortest independent vectors problem (SIVP). This improves the n O(n) running time of the best previously known algorithms for CVP [R. Kannan, and gives a deterministic and asymptotically faster alternative to the 2 O(n) -time (and space) randomized algorithm for SVP of Ajtai, Kumar, and Sivakumar [Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, 2001, pp. 266-275]. The core of our algorithm is a new method to solve the closest vector problem with preprocessing (CVPP) that uses the Voronoi cell of the lattice (described as intersection of half-spaces) as the result of the preprocessing function. A direct consequence of our results is a derandomization of the best current polynomial time approximation algorithms for SVP and CVP, achieving a 2 O(n log log n/ log n) approximation factor. Introduction.A d-dimensional lattice Λ is a discrete subgroup of the Euclidean space R d , and is customarily represented as the set of all integer linear combinations of n ≤ d basis vectors B = [b 1 , . . . , b n ] ∈ R d×n . There are many famous algorithmic problems on point lattices, the most important of which are as follows:• The shortest vector problem (SVP): given a basis B, find a shortest nonzero vector in the lattice generated by B. • The closest vector problem (CVP): given a basis B and a target vector t ∈ R d , find a lattice vector generated by B that is closest to t. • The shortest independent vectors problem (SIVP): given a basis B, find n linearly independent lattice vectors in the lattice generated by B that are as short as possible. 1 Besides being classic mathematical problems in the study of the geometry of numbers [16], these problems play an important role in many computer science and communication theory applications. SVP and CVP have been used to solve many landmark algorithmic problems in theoretical computer science, like integer programming [41,36], factoring polynomials over the rationals [40], checking the solvability by radicals [39],

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“…Subsequently, approximation hardness for the gap version of CVPP (i.e. approximately deciding the distance of the target) was shown in [11,27,4], culminating in a hardness factor of 2 log 1−ε n for any ε > 0 [17] under the assumption that NP is not in randomized quasi-polynomial time. On the positive side, polynomial time algorithms for the approximate search version of CVPP were studied (implicitly) in [5,18], where the current best approximation factor O(n/ log n) was recently achieved in [7].…”

Section: Introductionmentioning

confidence: 99%

Short Paths on the Voronoi Graph and Closest Vector Problem with Preprocessing

Bonifas¹,

Dadush²

2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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Improving on the Voronoi cell based techniques of [28, 24], we give a Las Vegas O(2 n ) expected time and space algorithm for CVPP (the preprocessing version of the Closest Vector Problem, CVP). This improves on the O(4 n ) deterministic runtime of the Micciancio Voulgaris algorithm [24] (henceforth MV) for CVPP 1 at the cost of a polynomial amount of randomness (which only affects runtime, not correctness).As in MV, our algorithm proceeds by computing a short path on the Voronoi graph of the lattice, where lattice points are adjacent if their Voronoi cells share a common facet, from the origin to a closest lattice vector. Our main technical contribution is a randomized procedure that, given the Voronoi relevant vectors of a lattice -the lattice vectors inducing facets of the Voronoi cell -as preprocessing, and any "close enough" lattice point to the target, computes a path to a closest lattice vector of expected polynomial size. This improves on the O(2 n ) path length given by the MV algorithm. Furthermore, as in MV, each edge of the path can be computed using a single iteration over the Voronoi relevant vectors.As a byproduct of our work, we also give an optimal relationship between geometric and path distance on the Voronoi graph, which we believe to be of independent interest.

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2 ^{log1-ε
n} hardness for the closest vector problem with preprocessing

Cited by 6 publications

References 24 publications

Universal Factor Graphs

Universal Factor Graphs

A Deterministic Single Exponential Time Algorithm for Most Lattice Problems Based on Voronoi Cell Computations

Short Paths on the Voronoi Graph and Closest Vector Problem with Preprocessing

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2 log1-ε n hardness for the closest vector problem with preprocessing

Cited by 6 publications

References 24 publications

Universal Factor Graphs

Universal Factor Graphs

A Deterministic Single Exponential Time Algorithm for Most Lattice Problems Based on Voronoi Cell Computations

Short Paths on the Voronoi Graph and Closest Vector Problem with Preprocessing

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2 ^{log1-ε
n} hardness for the closest vector problem with preprocessing