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18
PolynomialTime Approximation Schemes for Geometric Graphs
, 2001
"... A disk graph is the intersection graph of a set of disks with arbitrary diameters in the plane. For the case that the disk representation is given, we present polynomialtime approximation schemes (PTASs) for the maximum weight independent set problem (selecting disjoint disks of maximum total weigh ..."
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Cited by 102 (5 self)
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A disk graph is the intersection graph of a set of disks with arbitrary diameters in the plane. For the case that the disk representation is given, we present polynomialtime approximation schemes (PTASs) for the maximum weight independent set problem (selecting disjoint disks of maximum total weight) and for the minimum weight vertex cover problem in disk graphs. These are the first known PTASs for NPhard optimization problems on disk graphs. They are based on a novel recursive subdivision of the plane that allows applying a shifting strategy on different levels simultaneously, so that a dynamic programming approach becomes feasible. The PTASs for disk graphs represent a common generalization of previous results for planar graphs and unit disk graphs. They can be extended to intersections graphs of other "disklike" geometric objects (such as squares or regular polygons), also in higher dimensions.
Geometric Separation and Exact Solutions for the Parameterized Independent Set Problem on Disk Graphs
, 2002
"... We consider the parameterized problem, whether for a given set D of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k nonintersecting disks. We expose an algorithm running in time n , that isto our knowledgethe rst algorithm for this problem with running t ..."
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Cited by 23 (2 self)
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We consider the parameterized problem, whether for a given set D of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k nonintersecting disks. We expose an algorithm running in time n , that isto our knowledgethe rst algorithm for this problem with running time bounded by an exponential with a sublinear exponent. For precision disk graphs of bounded radius ratio, we show that the problem is xed parameter tractable with respect to parameter k.
Efficient Approximation Schemes for Geometric Problems?
, 2005
"... An EPTAS (efficient PTAS) is an approximation scheme where ǫ does not appear in the exponent of n, i.e., the running time is f(ǫ) ·nc. We use parameterized complexity to investigate the possibility of improving the known approximation schemes for certain geometric problems to EPTAS. Answering an op ..."
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Cited by 19 (3 self)
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An EPTAS (efficient PTAS) is an approximation scheme where ǫ does not appear in the exponent of n, i.e., the running time is f(ǫ) ·nc. We use parameterized complexity to investigate the possibility of improving the known approximation schemes for certain geometric problems to EPTAS. Answering an open question of Alber and Fiala [2], we show that Maximum Independent Set is W[1]complete for the intersection graphs of unit disks and axisparallel unit squares in the plane. A standard consequence of this result is that the nO(1/ǫ) time PTAS of Hunt et al. [11] for Maximum Independent Set on unit disk graphs cannot be improved to an EPTAS. Similar results are obtained for the problem of covering points with squares.
On the optimality of planar and geometric approximation schemes
"... We show for several planar and geometric problems that the best known approximation schemes are essentially optimal with respect to the dependence on ǫ. For example, we show that the 2O(1/ǫ) · n time approximation schemes for planar MAXIMUM INDEPENDENT SET and for TSP on a metric defined by a plan ..."
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Cited by 17 (5 self)
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We show for several planar and geometric problems that the best known approximation schemes are essentially optimal with respect to the dependence on ǫ. For example, we show that the 2O(1/ǫ) · n time approximation schemes for planar MAXIMUM INDEPENDENT SET and for TSP on a metric defined by a planar graph are essentially optimal: if there is a δ> 0 such that any of these problems admits a 2O((1/ǫ)1−δ) O(1) n time PTAS, then the Exponential Time Hypothesis (ETH) fails. It is known that MAXIMUM INDEPENDENT SET on unit disk graphs and the planar logic problems MPSAT, TMIN, TMAX admit nO(1/ǫ) time approximation schemes. We show that they are optimal in the sense that if there is a δ> 0 such that any of these problems admits a 2 (1/ǫ)O(1) nO((1/ǫ)1−δ) time PTAS, then ETH fails.
Independent Set of Intersection Graphs of Convex Objects in 2D
"... The intersection graph of a set of geometric objects is defined as a graph G = (S; E) in which there is an edge between two nodes si; sj 2 S if si " sj 6 =;. The problem of computing a maximum independent set in the intersection graph of a set of objects is known to be N Pcomplete for mos ..."
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Cited by 17 (1 self)
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The intersection graph of a set of geometric objects is defined as a graph G = (S; E) in which there is an edge between two nodes si; sj 2 S if si &quot; sj 6 =;. The problem of computing a maximum independent set in the intersection graph of a set of objects is known to be N Pcomplete for most cases in two and higher dimensions. We present approximation algorithms for computing a maximum independent set of intersection graphs of convex objects in R 2. Specifically, given (i) a set of n line segments in the plane with maximum independent set of size ff, we present algorithms that find an independent set of size at least (ff=(2 log(2n=ff)))
Computing the Independence Number of Intersection Graphs
"... Computing the maximum number of disjoint elements in a collection C of geometric objects is a classical problem in computational geometry with applications ranging from frequency assignment in cellular networks to map labeling in computational cartography. The problem is equivalent to finding the in ..."
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Cited by 11 (0 self)
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Computing the maximum number of disjoint elements in a collection C of geometric objects is a classical problem in computational geometry with applications ranging from frequency assignment in cellular networks to map labeling in computational cartography. The problem is equivalent to finding the independence number, α(GC), of the intersection graph GC of C, obtained by connecting two elements of C with an edge if and only if their intersection is nonempty. This is known to be an NPhard task even for systems of segments in the plane with at most two different slopes. The best known polynomial time approximation algorithm for systems of arbitrary segments is due to Agarwal and Mustafa, and returns in the worst case an n 1/2+o(1)approximation for α. Using extensions of the LiptonTarjan separator theorem, we improve this result and present, for every ɛ> 0, a polynomial time algorithm for computing α(GC) with approximation ratio at most n ɛ. In contrast, for general graphs, for any ɛ> 0 it is NPhard to approximate the independence number within a factor of n 1−ɛ. We also give a subexponential time exact algorithm for computing the independence number of intersection graphs of arcwise connected sets in the plane. 1
On grids in topological graphs
 Proc. 25nd ACM Symp. on Computational Geometry (SoCG
, 2009
"... A topological graph is a graph drawn in the plane with vertices represented by points and edges as arcs connecting its vertices. A kgrid in a topological graph is a pair of edge subsets, each of size k, such that every edge in one subset crosses every edge in the other subset. It is known that for ..."
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Cited by 9 (6 self)
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A topological graph is a graph drawn in the plane with vertices represented by points and edges as arcs connecting its vertices. A kgrid in a topological graph is a pair of edge subsets, each of size k, such that every edge in one subset crosses every edge in the other subset. It is known that for a fixed constant k, every nvertex topological graph with no kgrid has O(n) edges. We conjecture that this remains true even when: (1) considering grids with distinct vertices; or (2) all edges are straightline segments and the edges within each subset of the grid are required to be pairwise disjoint. These conjectures are shown to be true apart from log ∗ n and log 2 n factors, respectively. We also settle the conjectures for some special cases, including the second conjecture for convex geometric graphs. This result follows from a stronger statement that generalizes the celebrated MarcusTardos Theorem on excluded patterns in 01 matrices. 1
Independence and Coloring Problems on Intersection Graphs of Disks
, 2001
"... This paper surveys online and approximation algorithms for the maximum independent set and coloring problems on intersection graphs of disks. As a new result, it is shown that no deterministic online algorithm can achieve competitive ratio better than (log n) for disk graphs and for square g ..."
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Cited by 8 (2 self)
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This paper surveys online and approximation algorithms for the maximum independent set and coloring problems on intersection graphs of disks. As a new result, it is shown that no deterministic online algorithm can achieve competitive ratio better than (log n) for disk graphs and for square graphs with n vertices, even if the geometric representation is given as part of the input. Furthermore, it is proved that the standard First t heuristic achieves competitive ratio O(log n) for disk graphs and for square graphs and is thus best possible.
Approximation Algorithms for Intersection Graphs
, 2009
"... We introduce three new complexity parameters that in some sense measure how chordallike a graph is. The similarity to chordal graphs is used to construct simple polynomialtime approximation algorithms with constant approximation ratio for many NPhard problems, when restricted to graphs for which ..."
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Cited by 8 (0 self)
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We introduce three new complexity parameters that in some sense measure how chordallike a graph is. The similarity to chordal graphs is used to construct simple polynomialtime approximation algorithms with constant approximation ratio for many NPhard problems, when restricted to graphs for which at least one of our new complexity parameters is bounded by a constant. As applications we present approximation algorithms with constant approximation ratio for maximum weighted independent set, minimum (independent) dominating set, minimum vertex coloring, maximum weighted clique, and minimum clique partition for large classes of intersection graphs.
On distance constrained labeling of disk graphs
, 2004
"... A disk graph is the intersection graph of a set of disks in the plane. For a ktuple (p1,..., pk) of positive integers, a distance constrained labeling of a graph G is an assignment of labels to the vertices of G such that the labels of any pair of vertices at graph distance i in G differ by at leas ..."
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Cited by 8 (2 self)
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A disk graph is the intersection graph of a set of disks in the plane. For a ktuple (p1,..., pk) of positive integers, a distance constrained labeling of a graph G is an assignment of labels to the vertices of G such that the labels of any pair of vertices at graph distance i in G differ by at least pi, for i = 1,..., k. In the case when k = 1 and p1 = 1, this gives a traditional coloring of G. We propose and analyze several online and offline labeling algorithms for the class of disk graphs.