7,075 research outputs found
Search-to-Decision Reductions for Lattice Problems with Approximation Factors (Slightly) Greater Than One
We show the first dimension-preserving search-to-decision reductions for
approximate SVP and CVP. In particular, for any ,
we obtain an efficient dimension-preserving reduction from -SVP to -GapSVP and an efficient dimension-preserving reduction
from -CVP to -GapCVP. These results generalize the known
equivalences of the search and decision versions of these problems in the exact
case when . For SVP, we actually obtain something slightly stronger
than a search-to-decision reduction---we reduce -SVP to
-unique SVP, a potentially easier problem than -GapSVP.Comment: Updated to acknowledge additional prior wor
Inapproximability of Combinatorial Optimization Problems
We survey results on the hardness of approximating combinatorial optimization
problems
On the Closest Vector Problem with a Distance Guarantee
We present a substantially more efficient variant, both in terms of running
time and size of preprocessing advice, of the algorithm by Liu, Lyubashevsky,
and Micciancio for solving CVPP (the preprocessing version of the Closest
Vector Problem, CVP) with a distance guarantee. For instance, for any , our algorithm finds the (unique) closest lattice point for any target
point whose distance from the lattice is at most times the length of
the shortest nonzero lattice vector, requires as preprocessing advice only vectors, and runs in
time .
As our second main contribution, we present reductions showing that it
suffices to solve CVP, both in its plain and preprocessing versions, when the
input target point is within some bounded distance of the lattice. The
reductions are based on ideas due to Kannan and a recent sparsification
technique due to Dadush and Kun. Combining our reductions with the LLM
algorithm gives an approximation factor of for search
CVPP, improving on the previous best of due to Lagarias, Lenstra,
and Schnorr. When combined with our improved algorithm we obtain, somewhat
surprisingly, that only O(n) vectors of preprocessing advice are sufficient to
solve CVPP with (the only slightly worse) approximation factor of O(n).Comment: An early version of the paper was titled "On Bounded Distance
Decoding and the Closest Vector Problem with Preprocessing". Conference on
Computational Complexity (2014
Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling
A distance labeling scheme is an assignment of bit-labels to the vertices of
an undirected, unweighted graph such that the distance between any pair of
vertices can be decoded solely from their labels. An important class of
distance labeling schemes is that of hub labelings, where a node
stores its distance to the so-called hubs , chosen so that for
any there is belonging to some shortest
path. Notice that for most existing graph classes, the best distance labelling
constructions existing use at some point a hub labeling scheme at least as a
key building block. Our interest lies in hub labelings of sparse graphs, i.e.,
those with , for which we show a lowerbound of
for the average size of the hubsets.
Additionally, we show a hub-labeling construction for sparse graphs of average
size for some , where is the
so-called Ruzsa-Szemer{\'e}di function, linked to structure of induced
matchings in dense graphs. This implies that further improving the lower bound
on hub labeling size to would require a
breakthrough in the study of lower bounds on , which have resisted
substantial improvement in the last 70 years. For general distance labeling of
sparse graphs, we show a lowerbound of , where is the communication complexity of the
Sum-Index problem over . Our results suggest that the best achievable
hub-label size and distance-label size in sparse graphs may be
for some
On the Lattice Distortion Problem
We introduce and study the \emph{Lattice Distortion Problem} (LDP). LDP asks
how "similar" two lattices are. I.e., what is the minimal distortion of a
linear bijection between the two lattices? LDP generalizes the Lattice
Isomorphism Problem (the lattice analogue of Graph Isomorphism), which simply
asks whether the minimal distortion is one.
As our first contribution, we show that the distortion between any two
lattices is approximated up to a factor by a simple function of
their successive minima. Our methods are constructive, allowing us to compute
low-distortion mappings that are within a factor
of optimal in polynomial time and within a factor of optimal in
singly exponential time. Our algorithms rely on a notion of basis reduction
introduced by Seysen (Combinatorica 1993), which we show is intimately related
to lattice distortion. Lastly, we show that LDP is NP-hard to approximate to
within any constant factor (under randomized reductions), by a reduction from
the Shortest Vector Problem.Comment: This is the full version of a paper that appeared in ESA 201
Decoding by Embedding: Correct Decoding Radius and DMT Optimality
The closest vector problem (CVP) and shortest (nonzero) vector problem (SVP)
are the core algorithmic problems on Euclidean lattices. They are central to
the applications of lattices in many problems of communications and
cryptography. Kannan's \emph{embedding technique} is a powerful technique for
solving the approximate CVP, yet its remarkable practical performance is not
well understood. In this paper, the embedding technique is analyzed from a
\emph{bounded distance decoding} (BDD) viewpoint. We present two complementary
analyses of the embedding technique: We establish a reduction from BDD to
Hermite SVP (via unique SVP), which can be used along with any Hermite SVP
solver (including, among others, the Lenstra, Lenstra and Lov\'asz (LLL)
algorithm), and show that, in the special case of LLL, it performs at least as
well as Babai's nearest plane algorithm (LLL-aided SIC). The former analysis
helps to explain the folklore practical observation that unique SVP is easier
than standard approximate SVP. It is proven that when the LLL algorithm is
employed, the embedding technique can solve the CVP provided that the noise
norm is smaller than a decoding radius , where
is the minimum distance of the lattice, and . This
substantially improves the previously best known correct decoding bound . Focusing on the applications of BDD to decoding of
multiple-input multiple-output (MIMO) systems, we also prove that BDD of the
regularized lattice is optimal in terms of the diversity-multiplexing gain
tradeoff (DMT), and propose practical variants of embedding decoding which
require no knowledge of the minimum distance of the lattice and/or further
improve the error performance.Comment: To appear in IEEE Transactions on Information Theor
Hardness of Graph Pricing through Generalized Max-Dicut
The Graph Pricing problem is among the fundamental problems whose
approximability is not well-understood. While there is a simple combinatorial
1/4-approximation algorithm, the best hardness result remains at 1/2 assuming
the Unique Games Conjecture (UGC). We show that it is NP-hard to approximate
within a factor better than 1/4 under the UGC, so that the simple combinatorial
algorithm might be the best possible. We also prove that for any , there exists such that the integrality gap of
-rounds of the Sherali-Adams hierarchy of linear programming for
Graph Pricing is at most 1/2 + .
This work is based on the effort to view the Graph Pricing problem as a
Constraint Satisfaction Problem (CSP) simpler than the standard and complicated
formulation. We propose the problem called Generalized Max-Dicut(), which
has a domain size for every . Generalized Max-Dicut(1) is
well-known Max-Dicut. There is an approximation-preserving reduction from
Generalized Max-Dicut on directed acyclic graphs (DAGs) to Graph Pricing, and
both our results are achieved through this reduction. Besides its connection to
Graph Pricing, the hardness of Generalized Max-Dicut is interesting in its own
right since in most arity two CSPs studied in the literature, SDP-based
algorithms perform better than LP-based or combinatorial algorithms --- for
this arity two CSP, a simple combinatorial algorithm does the best.Comment: 28 page
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