44,081 research outputs found
The Hardness of Approximation of Euclidean k-means
The Euclidean -means problem is a classical problem that has been
extensively studied in the theoretical computer science, machine learning and
the computational geometry communities. In this problem, we are given a set of
points in Euclidean space , and the goal is to choose centers in
so that the sum of squared distances of each point to its nearest center
is minimized. The best approximation algorithms for this problem include a
polynomial time constant factor approximation for general and a
-approximation which runs in time . At
the other extreme, the only known computational complexity result for this
problem is NP-hardness [ADHP'09]. The main difficulty in obtaining hardness
results stems from the Euclidean nature of the problem, and the fact that any
point in can be a potential center. This gap in understanding left open
the intriguing possibility that the problem might admit a PTAS for all .
In this paper we provide the first hardness of approximation for the
Euclidean -means problem. Concretely, we show that there exists a constant
such that it is NP-hard to approximate the -means objective
to within a factor of . We show this via an efficient reduction
from the vertex cover problem on triangle-free graphs: given a triangle-free
graph, the goal is to choose the fewest number of vertices which are incident
on all the edges. Additionally, we give a proof that the current best hardness
results for vertex cover can be carried over to triangle-free graphs. To show
this we transform , a known hard vertex cover instance, by taking a graph
product with a suitably chosen graph , and showing that the size of the
(normalized) maximum independent set is almost exactly preserved in the product
graph using a spectral analysis, which might be of independent interest
A Novel Approach to Finding Near-Cliques: The Triangle-Densest Subgraph Problem
Many graph mining applications rely on detecting subgraphs which are
near-cliques. There exists a dichotomy between the results in the existing work
related to this problem: on the one hand the densest subgraph problem (DSP)
which maximizes the average degree over all subgraphs is solvable in polynomial
time but for many networks fails to find subgraphs which are near-cliques. On
the other hand, formulations that are geared towards finding near-cliques are
NP-hard and frequently inapproximable due to connections with the Maximum
Clique problem.
In this work, we propose a formulation which combines the best of both
worlds: it is solvable in polynomial time and finds near-cliques when the DSP
fails. Surprisingly, our formulation is a simple variation of the DSP.
Specifically, we define the triangle densest subgraph problem (TDSP): given
, find a subset of vertices such that , where is the number of triangles induced
by the set . We provide various exact and approximation algorithms which the
solve the TDSP efficiently. Furthermore, we show how our algorithms adapt to
the more general problem of maximizing the -clique average density. Finally,
we provide empirical evidence that the TDSP should be used whenever the output
of the DSP fails to output a near-clique.Comment: 42 page
Approximation Algorithms for Multi-Criteria Traveling Salesman Problems
In multi-criteria optimization problems, several objective functions have to
be optimized. Since the different objective functions are usually in conflict
with each other, one cannot consider only one particular solution as the
optimal solution. Instead, the aim is to compute a so-called Pareto curve of
solutions. Since Pareto curves cannot be computed efficiently in general, we
have to be content with approximations to them.
We design a deterministic polynomial-time algorithm for multi-criteria
g-metric STSP that computes (min{1 +g, 2g^2/(2g^2 -2g +1)} + eps)-approximate
Pareto curves for all 1/2<=g<=1. In particular, we obtain a
(2+eps)-approximation for multi-criteria metric STSP. We also present two
randomized approximation algorithms for multi-criteria g-metric STSP that
achieve approximation ratios of (2g^3 +2g^2)/(3g^2 -2g +1) + eps and (1 +g)/(1
+3g -4g^2) + eps, respectively.
Moreover, we present randomized approximation algorithms for multi-criteria
g-metric ATSP (ratio 1/2 + g^3/(1 -3g^2) + eps) for g < 1/sqrt(3)), STSP with
weights 1 and 2 (ratio 4/3) and ATSP with weights 1 and 2 (ratio 3/2). To do
this, we design randomized approximation schemes for multi-criteria cycle cover
and graph factor problems.Comment: To appear in Algorithmica. A preliminary version has been presented
at the 4th Workshop on Approximation and Online Algorithms (WAOA 2006
Bilu-Linial Stable Instances of Max Cut and Minimum Multiway Cut
We investigate the notion of stability proposed by Bilu and Linial. We obtain
an exact polynomial-time algorithm for -stable Max Cut instances with
for some absolute constant . Our
algorithm is robust: it never returns an incorrect answer; if the instance is
-stable, it finds the maximum cut, otherwise, it either finds the
maximum cut or certifies that the instance is not -stable. We prove
that there is no robust polynomial-time algorithm for -stable instances
of Max Cut when , where is the best
approximation factor for Sparsest Cut with non-uniform demands.
Our algorithm is based on semidefinite programming. We show that the standard
SDP relaxation for Max Cut (with triangle inequalities) is integral
if , where
is the least distortion with which every point metric space of negative
type embeds into . On the negative side, we show that the SDP
relaxation is not integral when .
Moreover, there is no tractable convex relaxation for -stable instances
of Max Cut when . That suggests that solving
-stable instances with might be difficult or
impossible.
Our results significantly improve previously known results. The best
previously known algorithm for -stable instances of Max Cut required
that (for some ) [Bilu, Daniely, Linial, and
Saks]. No hardness results were known for the problem. Additionally, we present
an algorithm for 4-stable instances of Minimum Multiway Cut. We also study a
relaxed notion of weak stability.Comment: 24 page
A QPTAS for Maximum Weight Independent Set of Polygons with Polylogarithmically Many Vertices
The Maximum Weight Independent Set of Polygons problem is a fundamental
problem in computational geometry. Given a set of weighted polygons in the
2-dimensional plane, the goal is to find a set of pairwise non-overlapping
polygons with maximum total weight. Due to its wide range of applications, the
MWISP problem and its special cases have been extensively studied both in the
approximation algorithms and the computational geometry community. Despite a
lot of research, its general case is not well-understood. Currently the best
known polynomial time algorithm achieves an approximation ratio of n^(epsilon)
[Fox and Pach, SODA 2011], and it is not even clear whether the problem is
APX-hard. We present a (1+epsilon)-approximation algorithm, assuming that each
polygon in the input has at most a polylogarithmic number of vertices. Our
algorithm has quasi-polynomial running time.
We use a recently introduced framework for approximating maximum weight
independent set in geometric intersection graphs. The framework has been used
to construct a QPTAS in the much simpler case of axis-parallel rectangles. We
extend it in two ways, to adapt it to our much more general setting. First, we
show that its technical core can be reduced to the case when all input polygons
are triangles. Secondly, we replace its key technical ingredient which is a
method to partition the plane using only few edges such that the objects
stemming from the optimal solution are evenly distributed among the resulting
faces and each object is intersected only a few times. Our new procedure for
this task is not more complex than the original one, and it can handle the
arising difficulties due to the arbitrary angles of the polygons. Note that
already this obstacle makes the known analysis for the above framework fail.
Also, in general it is not well understood how to handle this difficulty by
efficient approximation algorithms
Exact asymptotics of the optimal Lp-error of asymmetric linear spline approximation
In this paper we study the best asymmetric (sometimes also called penalized
or sign-sensitive) approximation in the metrics of the space , , of functions with nonnegative
Hessian by piecewise linear splines , generated by given
triangulations with elements. We find the exact asymptotic
behavior of optimal (over triangulations and splines error of such approximation as
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