5,402 research outputs found
On the Complexity of Local Distributed Graph Problems
This paper is centered on the complexity of graph problems in the
well-studied LOCAL model of distributed computing, introduced by Linial [FOCS
'87]. It is widely known that for many of the classic distributed graph
problems (including maximal independent set (MIS) and -vertex
coloring), the randomized complexity is at most polylogarithmic in the size
of the network, while the best deterministic complexity is typically
. Understanding and narrowing down this exponential gap
is considered to be one of the central long-standing open questions in the area
of distributed graph algorithms. We investigate the problem by introducing a
complexity-theoretic framework that allows us to shed some light on the role of
randomness in the LOCAL model. We define the SLOCAL model as a sequential
version of the LOCAL model. Our framework allows us to prove completeness
results with respect to the class of problems which can be solved efficiently
in the SLOCAL model, implying that if any of the complete problems can be
solved deterministically in rounds in the LOCAL model, we can
deterministically solve all efficient SLOCAL-problems (including MIS and
-coloring) in rounds in the LOCAL model. We show
that a rather rudimentary looking graph coloring problem is complete in the
above sense: Color the nodes of a graph with colors red and blue such that each
node of sufficiently large polylogarithmic degree has at least one neighbor of
each color. The problem admits a trivial zero-round randomized solution. The
result can be viewed as showing that the only obstacle to getting efficient
determinstic algorithms in the LOCAL model is an efficient algorithm to
approximately round fractional values into integer values
The parameterized complexity of some geometric problems in unbounded dimension
We study the parameterized complexity of the following fundamental geometric
problems with respect to the dimension : i) Given points in \Rd,
compute their minimum enclosing cylinder. ii) Given two -point sets in
\Rd, decide whether they can be separated by two hyperplanes. iii) Given a
system of linear inequalities with variables, find a maximum-size
feasible subsystem. We show that (the decision versions of) all these problems
are W[1]-hard when parameterized by the dimension . %and hence not solvable
in time, for any computable function and constant
%(unless FPT=W[1]). Our reductions also give a -time lower bound
(under the Exponential Time Hypothesis)
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
The Bane of Low-Dimensionality Clustering
In this paper, we give a conditional lower bound of on
running time for the classic k-median and k-means clustering objectives (where
n is the size of the input), even in low-dimensional Euclidean space of
dimension four, assuming the Exponential Time Hypothesis (ETH). We also
consider k-median (and k-means) with penalties where each point need not be
assigned to a center, in which case it must pay a penalty, and extend our lower
bound to at least three-dimensional Euclidean space.
This stands in stark contrast to many other geometric problems such as the
traveling salesman problem, or computing an independent set of unit spheres.
While these problems benefit from the so-called (limited) blessing of
dimensionality, as they can be solved in time or
in d dimensions, our work shows that widely-used clustering
objectives have a lower bound of , even in dimension four.
We complete the picture by considering the two-dimensional case: we show that
there is no algorithm that solves the penalized version in time less than
, and provide a matching upper bound of .
The main tool we use to establish these lower bounds is the placement of
points on the moment curve, which takes its inspiration from constructions of
point sets yielding Delaunay complexes of high complexity
New (and Old) Proof Systems for Lattice Problems
We continue the study of statistical zero-knowledge (SZK)
proofs, both interactive and noninteractive, for computational
problems on point lattices. We are particularly interested in the
problem GapSPP of approximating the -smoothing
parameter (for some ) of an -dimensional
lattice. The smoothing parameter is a key quantity in the study of
lattices, and GapSPP has been emerging as a core problem in
lattice-based cryptography, e.g., in worst-case to average-case
reductions.
We show that GapSPP admits SZK proofs for *remarkably low*
approximation factors, improving on prior work by up to
roughly . Specifically:
-- There is a *noninteractive* SZK proof for
-approximate GapSPP. Moreover,
for any negligible and a larger approximation factor
, there is such a proof with an
*efficient prover*.
-- There is an (interactive) SZK proof with an efficient prover for
-approximate
coGapSPP. We show this by proving that
-approximate GapSPP is in coNP.
In addition, we give an (interactive) SZK proof with an efficient
prover for approximating the lattice *covering radius* to within
an factor, improving upon the prior best factor of
Inapproximability of the independent set polynomial in the complex plane
We study the complexity of approximating the independent set polynomial
of a graph with maximum degree when the activity
is a complex number.
This problem is already well understood when is real using
connections to the -regular tree . The key concept in that case is
the "occupation ratio" of the tree . This ratio is the contribution to
from independent sets containing the root of the tree, divided
by itself. If is such that the occupation ratio
converges to a limit, as the height of grows, then there is an FPTAS for
approximating on a graph with maximum degree .
Otherwise, the approximation problem is NP-hard.
Unsurprisingly, the case where is complex is more challenging.
Peters and Regts identified the complex values of for which the
occupation ratio of the -regular tree converges. These values carve a
cardioid-shaped region in the complex plane. Motivated by the
picture in the real case, they asked whether marks the true
approximability threshold for general complex values .
Our main result shows that for every outside of ,
the problem of approximating on graphs with maximum degree
at most is indeed NP-hard. In fact, when is outside of
and is not a positive real number, we give the stronger result
that approximating is actually #P-hard. If is a
negative real number outside of , we show that it is #P-hard to
even decide whether , resolving in the affirmative a conjecture
of Harvey, Srivastava and Vondrak.
Our proof techniques are based around tools from complex analysis -
specifically the study of iterative multivariate rational maps
Flat currents modulo p in metric spaces and filling radius inequalities
We adapt the theory of currents in metric spaces, as developed by the
first-mentioned author in collaboration with B. Kirchheim, to currents with
coefficients in Z_p. Building on S. Wenger's work in the orientable case, we
obtain isoperimetric inequalities mod(p) in Banach spaces and we apply these
inequalities to provide a proof of Gromov's filling radius inequality (and
therefore also the systolic inequality) which applies to nonorientable
manifolds, as well. With this goal in mind, we use the Ekeland principle to
provide quasi-minimizers of the mass mod(p) in the homology class, and use the
isoperimetric inequality to give lower bounds on the growth of their mass in
balls.Comment: 31 pages, to appear in Commentarii Mathematici Helvetic
Shrinkwrapping and the taming of hyperbolic 3-manifolds
We introduce a new technique for finding CAT(-1) surfaces in hyperbolic
3-manifolds. We use this to show that a complete hyperbolic 3-manifold with
finitely generated fundamental group is geometrically and topologically tame.Comment: 60 pages, 7 figures; V3: incorporates referee's suggestions,
references update
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