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52
A characterization of the (natural) graph properties testable with onesided error
 Proc. of FOCS 2005
, 2005
"... The problem of characterizing all the testable graph properties is considered by many to be the most important open problem in the area of propertytesting. Our main result in this paper is a solution of an important special case of this general problem; Call a property tester oblivious if its decis ..."
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Cited by 111 (19 self)
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The problem of characterizing all the testable graph properties is considered by many to be the most important open problem in the area of propertytesting. Our main result in this paper is a solution of an important special case of this general problem; Call a property tester oblivious if its decisions are independent of the size of the input graph. We show that a graph property P has an oblivious onesided error tester, if and only if P is (almost) hereditary. We stress that any ”natural ” property that can be tested (either with onesided or with twosided error) can be tested by an oblivious tester. In particular, all the testers studied thus far in the literature were oblivious. Our main result can thus be considered as a precise characterization of the ”natural” graph properties, which are testable with onesided error. One of the main technical contributions of this paper is in showing that any hereditary graph property can be tested with onesided error. This general result contains as a special case all the previous results about testing graph properties with onesided error. These include the results of [20] and [5] about testing kcolorability, the characterization of [21] of the graphpartitioning problems that are testable with onesided error, the induced vertex colorability properties of [3], the induced edge colorability properties of [14], a transformation from twosided to onesided error testing [21], as well as a recent result about testing monotone graph properties [10]. More importantly, as a special case of our main result, we infer that some of the most well studied graph properties, both in graph theory and computer science, are testable with onesided error. Some of these properties are the well known graph properties of being Perfect, Chordal, Interval, Comparability, Permutation and more. None of these properties was previously known to be testable. 1
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 104 (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.
Maximum Independent Set of Rectangles
"... We study the Maximum Independent Set of Rectangles (MISR) problem: given a collection R of n axisparallel rectangles, find a maximumcardinality subset of disjoint rectangles. MISR is a special case of the classical Maximum Independent Set problem, where the input is restricted to intersection grap ..."
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Cited by 25 (0 self)
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We study the Maximum Independent Set of Rectangles (MISR) problem: given a collection R of n axisparallel rectangles, find a maximumcardinality subset of disjoint rectangles. MISR is a special case of the classical Maximum Independent Set problem, where the input is restricted to intersection graphs of axisparallel rectangles. Due to its many applications, ranging from map labeling to data mining, MISR has received a significant amount of attention from various research communities. Since the problem is NPhard, the main focus has been on the design of approximation algorithms. Several groups of researches have independently suggested O(log n)approximation algorithms for MISR, and this remained the best currently known approximation factor for the problem. The main result of our paper is an O(log log n)approximation algorithm for MISR. Our algorithm combines existing approaches for solving special cases of the problem, in which the input set of rectangles is restricted to containing specific intersection types, with new insights into the combinatorial structure of sets of intersecting rectangles in the plane. We also consider a generalization of MISR to higher dimensions, where rectangles are replaced by ddimensional hyperrectangles. Our results for MISR imply an O((log n) d−2 log log n)approximation algorithm for this problem, improving upon the best previously known O((log n) d−1)approximation.
Approximation Algorithms for Maximum Independent Set of PseudoDisks
, 2008
"... We present approximation algorithms for maximum independent set of pseudodisks in the plane, both in the weighted and unweighted cases. For the unweighted case, we prove that a local search algorithm yields a PTAS. For the weighted case, we suggest a novel rounding scheme based on an LP relaxation ..."
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Cited by 22 (4 self)
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We present approximation algorithms for maximum independent set of pseudodisks in the plane, both in the weighted and unweighted cases. For the unweighted case, we prove that a local search algorithm yields a PTAS. For the weighted case, we suggest a novel rounding scheme based on an LP relaxation of the problem, that leads to a constantfactor approximation. Most previous algorithms for maximum independent set (in geometric settings) relied on packing arguments that are not applicable in this case. As such, the analysis of both algorithms requires some new combinatorial ideas, which we believe to be of independent interest.
Optimization problems in multipleinterval graphs
 In Proceedings of the 18th annual Symposium On Discrete Algorithms (SODA
, 2007
"... Multipleinterval graphs are a natural generalization of interval graphs where each vertex may have more then one interval associated with it. We initiate the study of optimization problems in multipleinterval graphs by considering three classical problems: Minimum Vertex Cover, Minimum Dominating ..."
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Cited by 18 (5 self)
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Multipleinterval graphs are a natural generalization of interval graphs where each vertex may have more then one interval associated with it. We initiate the study of optimization problems in multipleinterval graphs by considering three classical problems: Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique. We describe applications for each one of these problems, and then proceed to discuss approximation algorithms for them. Our results can be summarized as follows: Let t be the number of intervals associated with each vertex in a given multipleinterval graph. For Minimum Vertex Cover, we give a (2 − 1/t)approximation algorithm which also works when a tinterval representation of our given graph is absent. Following this, we give a t 2approximation algorithm for Minimum Dominating Set which adapts well to more general variants of the problem. We then proceed to prove that Maximum Clique is NPhard already for 3interval graphs, and provide a (t 2 −t+ 1)/2approximation algorithm for general values of t ≥ 2, using bounds proven for the socalled transversal number of tinterval families.
A robust ptas for maximum weight independent sets in unit disk graphs
 In WG
, 2004
"... Abstract. A unit disk graph is the intersection graph of unit disks in the euclidean plane. We present a polynomialtime approximation scheme for the maximum weight independent set problem in unit disk graphs. In contrast to previously known approximation schemes, our approach does not require a geo ..."
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Cited by 18 (0 self)
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Abstract. A unit disk graph is the intersection graph of unit disks in the euclidean plane. We present a polynomialtime approximation scheme for the maximum weight independent set problem in unit disk graphs. In contrast to previously known approximation schemes, our approach does not require a geometric representation (specifying the coordinates of the disk centers). The approximation algorithm presented is robust in the sense that it accepts any graph as input and either returns a (1 + ε)approximate independent set or a certificate showing that the input graph is no unit disk graph. The algorithm can easily be extended to other families of intersection graphs of geometric objects. 1
Independent set of intersection graphs of convex objects in 2D
 in 2D. Comput. Geometry: Theory & Appls
, 2004
"... Abstract. The intersection graph of a set of geometric objects is defined as agraph 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 NPcomplete ..."
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Cited by 18 (1 self)
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Abstract. The intersection graph of a set of geometric objects is defined as agraph 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 NPcomplete for most casesin 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 a set of n line segments in the plane with maximum independent set of size ^, we present algorithms that find an independent set of size atleast ( i) (^=2 log(2n=^)) 1=2 in time O(n
Computationallyfeasible truthful auctions for convex bundles
 In RANDOM+APPROX
, 2004
"... In many economic settings, convex figures on the plane are for sale to a set of selfish agents. For example, we might want to sell an advertising space on a newspaper’s page or to sell a realestate lot. The selfish agents must be motivated to report their true values for the figures as well as to r ..."
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Cited by 12 (0 self)
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In many economic settings, convex figures on the plane are for sale to a set of selfish agents. For example, we might want to sell an advertising space on a newspaper’s page or to sell a realestate lot. The selfish agents must be motivated to report their true values for the figures as well as to report the true figures they are interested in (which are both private information). This is done by carefully designing a payment scheme for any chosen allocation scheme. Moreover, an approximation algorithm that runs in polynomial time should be used for guaranteeing a reasonable solution for the underlying NPcomplete problem. We present truthful mechanisms that guarantee a certain fraction of the social welfare, as a function of a measure R on the geometric diversity of the shapes: we define R as the ratio between the maximal diameter and the minimal width of figures that the agents bid for. We give the first approximation algorithm for packing arbitrary weighted compact convex figures. We use this algorithm, and variants of existing algorithms, to create polynomialtime truthful mechanisms that approximate the social welfare. For each mechanism, we show that it achieves the best approximate over all mechanisms of its kind. We also study different models of information (e.g., if only the agents ’ values are unknown, or both the values and the figures) and a discrete model where each player bids for a set of predefined building blocks that are contained in a convex figure. 1
A Note on Maximum Independent Sets in Rectangle Intersection Graphs
 Information Processing Letters
, 2003
"... Finding the maximum independent set in the intersection graph of n axisparallel rectangles is NPhard. We reexamine two known approximation results for this problem. For the case of rectangles of unit height, Agarwal, van Kreveld, and Suri (1997) gave a (1+1=k)factor algorithm with an O(n log ..."
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Cited by 11 (0 self)
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Finding the maximum independent set in the intersection graph of n axisparallel rectangles is NPhard. We reexamine two known approximation results for this problem. For the case of rectangles of unit height, Agarwal, van Kreveld, and Suri (1997) gave a (1+1=k)factor algorithm with an O(n log n + n ) time bound for any integer constant k 1; we describe a similar algorithm running in only O(n log n + n ) time, where n denotes the maximum number of rectangles a point can be in. For the general case, Berman, DasGupta, Muthukrishnan, and Ramaswami (2001) gave a dlog k nefactor algorithm with an O(n k+1 ) time bound for any integer constant k 2; we describe similar algorithms running in O(n log n + n ) and n O(k= log k) time.