10 research outputs found
Constant-Factor Approximation for TSP with Disks
We revisit the traveling salesman problem with neighborhoods (TSPN) and
present the first constant-ratio approximation for disks in the plane: Given a
set of disks in the plane, a TSP tour whose length is at most times
the optimal can be computed in time that is polynomial in . Our result is
the first constant-ratio approximation for a class of planar convex bodies of
arbitrary size and arbitrary intersections. In order to achieve a
-approximation, we reduce the traveling salesman problem with disks, up
to constant factors, to a minimum weight hitting set problem in a geometric
hypergraph. The connection between TSPN and hitting sets in geometric
hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure
A PTAS for Euclidean TSP with Hyperplane Neighborhoods
In the Traveling Salesperson Problem with Neighborhoods (TSPN), we are given
a collection of geometric regions in some space. The goal is to output a tour
of minimum length that visits at least one point in each region. Even in the
Euclidean plane, TSPN is known to be APX-hard, which gives rise to studying
more tractable special cases of the problem. In this paper, we focus on the
fundamental special case of regions that are hyperplanes in the -dimensional
Euclidean space. This case contrasts the much-better understood case of
so-called fat regions.
While for an exact algorithm with running time is known,
settling the exact approximability of the problem for has been repeatedly
posed as an open question. To date, only an approximation algorithm with
guarantee exponential in is known, and NP-hardness remains open.
For arbitrary fixed , we develop a Polynomial Time Approximation Scheme
(PTAS) that works for both the tour and path version of the problem. Our
algorithm is based on approximating the convex hull of the optimal tour by a
convex polytope of bounded complexity. Such polytopes are represented as
solutions of a sophisticated LP formulation, which we combine with the
enumeration of crucial properties of the tour. As the approximation guarantee
approaches , our scheme adjusts the complexity of the considered polytopes
accordingly.
In the analysis of our approximation scheme, we show that our search space
includes a sufficiently good approximation of the optimum. To do so, we develop
a novel and general sparsification technique to transform an arbitrary convex
polytope into one with a constant number of vertices and, in turn, into one of
bounded complexity in the above sense. Hereby, we maintain important properties
of the polytope
The Traveling Salesman Problem: Low-Dimensionality Implies a Polynomial Time Approximation Scheme
The Traveling Salesman Problem (TSP) is among the most famous NP-hard
optimization problems. We design for this problem a randomized polynomial-time
algorithm that computes a (1+eps)-approximation to the optimal tour, for any
fixed eps>0, in TSP instances that form an arbitrary metric space with bounded
intrinsic dimension.
The celebrated results of Arora (A-98) and Mitchell (M-99) prove that the
above result holds in the special case of TSP in a fixed-dimensional Euclidean
space. Thus, our algorithm demonstrates that the algorithmic tractability of
metric TSP depends on the dimensionality of the space and not on its specific
geometry. This result resolves a problem that has been open since the
quasi-polynomial time algorithm of Talwar (T-04)
A constant-factor approximation algorithm for the k
) Avrim Blum R. Ravi y Santosh Vempala z Abstract Given an undirected graph with non-negative edge costs and an integer k, the k-MST problem is that of finding a tree of minimum cost on k nodes. This problem is known to be NP-hard. We present a simple approximation algorithm that finds a solution whose cost is less than 17 times the cost of the optimum. This improves upon previous performance ratios for this problem -- O( p k) due to Ravi et al., O(log 2 k) due to Awerbuch et al, and the previous best bound of O(log k) due to Rajagopalan and Vazirani. Given any 0 ! ff ! 1, we first present a bicriteria approximation algorithm that outputs a tree on p ffk vertices of total cost at most 2pL (1\Gammaff)k , where L is the cost of the optimal k-MST. The running time of the algorithm is O(n 2 log 2 n) on an n-node graph. We then show how to use this algorithm to derive a constant factor approximation algorithm for the k-MST problem. The main subroutine in our algorithm is ..
The traveling salesman problem for lines, balls and planes
We revisit the traveling salesman problem with neighborhoods (TSPN) and
propose several new approximation algorithms. These constitute either first
approximations (for hyperplanes, lines, and balls in , for ) or improvements over previous approximations achievable in comparable times
(for unit disks in the plane).
\smallskip (I) Given a set of hyperplanes in , a TSP tour
whose length is at most times the optimal can be computed in
time, when is constant.
\smallskip (II) Given a set of lines in , a TSP tour whose
length is at most times the optimal can be computed in polynomial
time for all .
\smallskip (III) Given a set of unit balls in , a TSP tour
whose length is at most times the optimal can be computed in polynomial
time, when is constant.Comment: 30 pages, 9 figures; final version to appear in ACM Transactions on
Algorithm
A QPTAS for TSP with Fat Weakly Disjoint Neighborhoods in Doubling Metrics
We consider the Traveling Salesman Problem with Neighborhoods (TSPN) in doubling metrics. The goal is to find a shortest tour that visits each of a collection of n subsets (regions or neighborhoods) in the underlying metric space. We give a QPTAS when the regions are what we call α-fat weakly disjoint. This notion combines the existing notions of diameter variation, fatness and disjointness for geometric objects and generalizes these notions to any arbitrary metric space. Intuitively, the regions can be grouped into a bounded number of types, where in each type, the regions have similar upper bounds for their diameters, and each such region can designate a point such that these points are far away from one another. Our result generalizes the PTAS for TSPN on the Euclidean plane by Mitchell [27] and the QPTAS for TSP on doubling metrics by Talwar [30]. We also observe that our techniques directly extend to a QPTAS for the Group Steiner Tree Problem on doubling metrics, with the same assumption on the groups
-Coresets for Clustering (with Outliers) in Doubling Metrics
We study the problem of constructing -coresets for the -clustering problem in a doubling metric . An -coreset
is a weighted subset with weight function , such that for any -subset , it holds that
.
We present an efficient algorithm that constructs an -coreset
for the -clustering problem in , where the size of the coreset
only depends on the parameters and the doubling dimension
. To the best of our knowledge, this is the first efficient
-coreset construction of size independent of for general
clustering problems in doubling metrics.
To this end, we establish the first relation between the doubling dimension
of and the shattering dimension (or VC-dimension) of the range space
induced by the distance . Such a relation was not known before, since one
can easily construct instances in which neither one can be bounded by (some
function of) the other. Surprisingly, we show that if we allow a small
-distortion of the distance function , and consider the
notion of -error probabilistic shattering dimension, we can prove an
upper bound of for the probabilistic shattering dimension for
even weighted doubling metrics. We believe this new relation is of independent
interest and may find other applications.
We also study the robust coresets and centroid sets in doubling metrics. Our
robust coreset construction leads to new results in clustering and property
testing, and the centroid sets can be used to accelerate the local search
algorithms for clustering problems.Comment: Appeared in FOCS 2018, this is the full versio