24,992 research outputs found
A Framework for Approximation Schemes on Disk Graphs
We initiate a systematic study of approximation schemes for fundamental
optimization problems on disk graphs, a common generalization of both planar
graphs and unit-disk graphs. Our main contribution is a general framework for
designing efficient polynomial-time approximation schemes (EPTASes) for
vertex-deletion problems on disk graphs, which results in EPTASes for many
problems including Vertex Cover, Feedback Vertex Set, Small Cycle Hitting (in
particular, Triangle Hitting), -Hitting for , Path
Deletion, Pathwidth -Deletion, Component Order Connectivity, Bounded Degree
Deletion, Pseudoforest Deletion, Finite-Type Component Deletion, etc. All
EPTASes obtained using our framework are robust in the sense that they do not
require a realization of the input graph. To the best of our knowledge, prior
to this work, the only problems known to admit (E)PTASes on disk graphs are
Maximum Clique, Independent Set, Dominating set, and Vertex Cover, among which
the existing PTAS [Erlebach et al., SICOMP'05] and EPTAS [Leeuwen, SWAT'06] for
Vertex Cover require a realization of the input disk graph (while ours does
not).
The core of our framework is a reduction for a broad class of (approximation)
vertex-deletion problems from (general) disk graphs to disk graphs of bounded
local radius, which is a new invariant of disk graphs introduced in this work.
Disk graphs of bounded local radius can be viewed as a mild generalization of
planar graphs, which preserves certain nice properties of planar graphs.
Specifically, we prove that disk graphs of bounded local radius admit the
Excluded Grid Minor property and have locally bounded treewidth. This allows
existing techniques for designing approximation schemes on planar graphs (e.g.,
bidimensionality and Baker's technique) to be directly applied to disk graphs
of bounded local radius
Algorithms for Unit-Disk Graphs and Related Problems
In this dissertation, we study algorithms for several problems on unit-disk graphs and related problems. The unit-disk graph can be viewed as an intersection graph of a set of congruent disks. Unit-disk graphs have been extensively studied due to many of their applications, e.g., modeling the topology of wireless sensor networks. Lots of problems on unit-disk graphs have been considered in the literature, such as shortest paths, clique, independent set, distance oracle, diameter, etc. Specifically, we study the following problems in this dissertation: L1 shortest paths in unit-disk graphs, reverse shortest paths in unit-disk graphs, minimum bottleneck moving spanning tree, unit-disk range reporting, distance selection, etc. We develop efficient algorithms for these problems and our results are either first-known solutions or somehow improve the previous work.
Given a set P of n points in the plane and a parameter r \u3e 0, a unit-disk graph G(P) can be defined using P as its vertex set and two points of P are connected by an edge if the distance between these two points is at most r. The weight of an edge is one in the unweighted case and is equal to the distance between the two endpoints in the weighted case. Note that the distance between two points can be measured by different metrics, e.g., L1 or L2 metric.
In the first problem of L1 shortest paths in unit-disk graphs, we are given a point set P and a source point s ∈ P, the problem is to find all shortest paths from s to all other vertices in the L1 weighted unit-disk graph defined on set P. We present an O(n log n) time algorithm, which matches the Ω(n log n)-time lower bound. In the second problem, we are given a set P of n points, parameters r, λ \u3e 0, and two points s and t of P, the goal is to compute the smallest r such that the shortest path length between s and t in the unit-disk graph with respect to set P and parameter r is at most λ. This problem can be defined in both unweighted and weighted cases. We propose an algorithm of O(⌊λ⌋ · n log n) time and another algorithm of O(n5/4 log7/4 n) time for the unweighted case. We also given an O(n5/4 log5/2 n) time algorithm for the weighted case. In the third problem, we are given a set P of n points that are moving in the plane, the problem is to compute a spanning tree for these moving points that does not change its combinatorial structure during the point movement such that the bottleneck weight of the spanning tree (i.e., the largest Euclidean length of all edges) during the whole movement is minimized. We present an algorithm that runs in O(n4/3 log3 n) time. The fourth problem is unit-disk range reporting in which we are given a set P of n points in the plane and a value r, we need to construct a data structure so that given any query disk of radius r, all points of P in the disk can be reported efficiently. We build a data structure of O(n) space in O(n log n) time that can answer each query in O(k + log n) time, where k is the output size. The time complexity of our algorithm is the same as the previous result but our approach is much simpler. Finally, for the problem of distance selection, we are given a set P of n points in the plane and an integer 1 ≤ k ≤ (n2), the distance selection problem is to find the k-th smallest interpoint distance among all pairs of points of p. We propose an algorithm that runs in O(n4/3 log n) time. Our techniques yield two algorithmic frameworks for solving geometric optimization problems.
Many algorithms and techniques developed in this dissertation are quite general and fundamental, and we believe they will find other applications in future
Optimization in Geometric Graphs: Complexity and Approximation
We consider several related problems arising in geometric graphs. In particular,
we investigate the computational complexity and approximability properties of several optimization problems in unit ball graphs and develop algorithms to find exact
and approximate solutions. In addition, we establish complexity-based theoretical
justifications for several greedy heuristics.
Unit ball graphs, which are defined in the three dimensional Euclidian space, have
several application areas such as computational geometry, facility location and, particularly, wireless communication networks. Efficient operation of wireless networks
involves several decision problems that can be reduced to well known optimization
problems in graph theory. For instance, the notion of a \virtual backbone" in a wire-
less network is strongly related to a minimum connected dominating set in its graph
theoretic representation.
Motivated by the vastness of application areas, we study several problems including maximum independent set, minimum vertex coloring, minimum clique partition,
max-cut and min-bisection. Although these problems have been widely studied in
the context of unit disk graphs, which are the two dimensional version of unit ball
graphs, there is no established result on the complexity and approximation status
for some of them in unit ball graphs. Furthermore, unit ball graphs can provide a
better representation of real networks since the nodes are deployed in the three dimensional space. We prove complexity results and propose solution procedures for
several problems using geometrical properties of these graphs.
We outline a matching-based branch and bound solution procedure for the maximum k-clique problem in unit disk graphs and demonstrate its effectiveness through
computational tests. We propose using minimum bottleneck connected dominating
set problem in order to determine the optimal transmission range of a wireless network that will ensure a certain size of "virtual backbone". We prove that this problem
is NP-hard in general graphs but solvable in polynomial time in unit disk and unit
ball graphs.
We also demonstrate work on theoretical foundations for simple greedy heuristics.
Particularly, similar to the notion of "best" approximation algorithms with respect to
their approximation ratios, we prove that several simple greedy heuristics are "best"
in the sense that it is NP-hard to recognize the gap between the greedy solution
and the optimal solution. We show results for several well known problems such as
maximum clique, maximum independent set, minimum vertex coloring and discuss
extensions of these results to a more general class of problems.
In addition, we propose a "worst-out" heuristic based on edge contractions for
the max-cut problem and provide analytical and experimental comparisons with a
well known "best-in" approach and its modified versions
Bidimensionality and Geometric Graphs
In this paper we use several of the key ideas from Bidimensionality to give a
new generic approach to design EPTASs and subexponential time parameterized
algorithms for problems on classes of graphs which are not minor closed, but
instead exhibit a geometric structure. In particular we present EPTASs and
subexponential time parameterized algorithms for Feedback Vertex Set, Vertex
Cover, Connected Vertex Cover, Diamond Hitting Set, on map graphs and unit disk
graphs, and for Cycle Packing and Minimum-Vertex Feedback Edge Set on unit disk
graphs. Our results are based on the recent decomposition theorems proved by
Fomin et al [SODA 2011], and our algorithms work directly on the input graph.
Thus it is not necessary to compute the geometric representations of the input
graph. To the best of our knowledge, these results are previously unknown, with
the exception of the EPTAS and a subexponential time parameterized algorithm on
unit disk graphs for Vertex Cover, which were obtained by Marx [ESA 2005] and
Alber and Fiala [J. Algorithms 2004], respectively.
We proceed to show that our approach can not be extended in its full
generality to more general classes of geometric graphs, such as intersection
graphs of unit balls in R^d, d >= 3. Specifically we prove that Feedback Vertex
Set on unit-ball graphs in R^3 neither admits PTASs unless P=NP, nor
subexponential time algorithms unless the Exponential Time Hypothesis fails.
Additionally, we show that the decomposition theorems which our approach is
based on fail for disk graphs and that therefore any extension of our results
to disk graphs would require new algorithmic ideas. On the other hand, we prove
that our EPTASs and subexponential time algorithms for Vertex Cover and
Connected Vertex Cover carry over both to disk graphs and to unit-ball graphs
in R^d for every fixed d
Efficient algorithms for optimization problems involving semi-algebraic range searching
We present a general technique, based on parametric search with some twist,
for solving a variety of optimization problems on a set of semi-algebraic
geometric objects of constant complexity. The common feature of these problems
is that they involve a `growth parameter' and a semi-algebraic predicate
of constant complexity on pairs of input objects, which depends
on and is monotone in . One then defines a graph whose edges are
all the pairs for which is true, and seeks the smallest
value of for which some monotone property holds for .
Problems that fit into this context include (i) the reverse shortest path
problem in unit-disk graphs, recently studied by Wang and Zhao, (ii) the same
problem for weighted unit-disk graphs, with a decision procedure recently
provided by Wang and Xue, (iii) extensions of these problems to three and
higher dimensions, (iv) the discrete Fr\'echet distance with one-sided
shortcuts in higher dimensions, extending the study by Ben Avraham et al., (v)
perfect matchings in intersection graphs: given, e.g., a set of fat ellipses of
roughly the same size, find the smallest value such that if we expand each
of the ellipses by , the resulting intersection graph contains a perfect
matching, (vi) generalized distance selection problems: given, e.g., a set of
disjoint segments, find the 'th smallest distance among the pairwise
distances determined by the segments, for a given (sufficiently small but
superlinear) parameter , and (vii) the maximum-height independent towers
problem, in which we want to erect vertical towers of maximum height over a
1.5-dimensional terrain so that no pair of tower tips are mutually visible.
We obtain significantly improved solutions for problems (i), (ii) and (vi),
and new efficient solutions to the other problems.Comment: Significantly generalized and with additional applications. Notice
the change in titl
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