292 research outputs found
Spanners for Geometric Intersection Graphs
Efficient algorithms are presented for constructing spanners in geometric
intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is
obtained using efficient partitioning of the space into hypercubes and solving
bichromatic closest pair problems. The spanner construction has almost
equivalent complexity to the construction of Euclidean minimum spanning trees.
The results are extended to arbitrary ball graphs with a sub-quadratic running
time.
For unit ball graphs, the spanners have a small separator decomposition which
can be used to obtain efficient algorithms for approximating proximity problems
like diameter and distance queries. The results on compressed quadtrees,
geometric graph separators, and diameter approximation might be of independent
interest.Comment: 16 pages, 5 figures, Late
Algorithms and complexity analyses for some combinational optimization problems
The main focus of this dissertation is on classical combinatorial optimization problems in two important areas: scheduling and network design.
In the area of scheduling, the main interest is in problems in the master-slave model. In this model, each machine is either a master machine or a slave machine. Each job is associated with a preprocessing task, a slave task and a postprocessing task that must be executed in this order. Each slave task has a dedicated slave machine. All the preprocessing and postprocessing tasks share a single master machine or the same set of master machines. A job may also have an arbitrary release time before which the preprocessing task is not available to be processed. The main objective in this dissertation is to minimize the total completion time or the makespan. Both the complexity and algorithmic issues of these problems are considered. It is shown that the problem of minimizing the total completion time is strongly NP-hard even under severe constraints. Various efficient algorithms are designed to minimize the total completion time under various scenarios.
In the area of network design, the survivable network design problems are studied first. The input for this problem is an undirected graph G = (V, E), a non-negative cost for each edge, and a nonnegative connectivity requirement ruv for every (unordered) pair of vertices &ruv. The goal is to find a minimum-cost subgraph in which each pair of vertices u,v is joined by at least ruv edge (vertex)-disjoint paths. A Polynomial Time Approximation Scheme (PTAS) is designed for the problem when the graph is Euclidean and the connectivity requirement of any point is at most 2. PTASs or Quasi-PTASs are also designed for 2-edge-connectivity problem and biconnectivity problem and their variations in unweighted or weighted planar graphs.
Next, the problem of constructing geometric fault-tolerant spanners with low cost and bounded maximum degree is considered. The first result shows that there is a greedy algorithm which constructs fault-tolerant spanners having asymptotically optimal bounds for both the maximum degree and the total cost at the same time. Then an efficient algorithm is developed which finds fault-tolerant spanners with asymptotically optimal bound for the maximum degree and almost optimal bound for the total cost
The changing face of innovation policy: implications for the Northern Ireland economy
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-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
Online Directed Spanners and Steiner Forests
We present online algorithms for directed spanners and Steiner forests. These
problems fall under the unifying framework of online covering linear
programming formulations, developed by Buchbinder and Naor (MOR, 34, 2009),
based on primal-dual techniques. Our results include the following:
For the pairwise spanner problem, in which the pairs of vertices to be
spanned arrive online, we present an efficient randomized
-competitive algorithm for graphs with general lengths,
where is the number of vertices. With uniform lengths, we give an efficient
randomized -competitive algorithm, and an
efficient deterministic -competitive algorithm,
where is the number of terminal pairs. These are the first online
algorithms for directed spanners. In the offline setting, the current best
approximation ratio with uniform lengths is ,
due to Chlamtac, Dinitz, Kortsarz, and Laekhanukit (TALG 2020).
For the directed Steiner forest problem with uniform costs, in which the
pairs of vertices to be connected arrive online, we present an efficient
randomized -competitive algorithm. The
state-of-the-art online algorithm for general costs is due to Chakrabarty, Ene,
Krishnaswamy, and Panigrahi (SICOMP 2018) and is -competitive. In the offline version, the current best approximation
ratio with uniform costs is , due to Abboud
and Bodwin (SODA 2018).
A small modification of the online covering framework by Buchbinder and Naor
implies a polynomial-time primal-dual approach with separation oracles, which a
priori might perform exponentially many calls. We convert the online spanner
problem and the online Steiner forest problem into online covering problems and
round in a problem-specific fashion
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